{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Red Emphasis" -1 262 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 263 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 264 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "More examples of evaluating limit s" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canad a" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 22.3.2007" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "load calculus procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 264 10 "calculus.m" }{TEXT -1 37 " conta ins the code for the procedure " }{TEXT 0 8 "eval_lim" }{TEXT -1 25 " \+ used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be re ad into a Maple session by a command similar to the one that follows, \+ where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/calculus.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "A procedure for evaluating limits algebraically: " }{TEXT 0 14 "eval_lim/Limit" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "eval_l im/Limit: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 18 "Calling Sequence:\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " eval_lim(L)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 23 2 " L" }{TEXT -1 77 " - a limit Limit( fx, x=a) given using the inert function \"Limit\".\n" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The pro cedure " }{TEXT 0 9 "evallimit" }{TEXT -1 31 " attempts to evaluate a \+ limit. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 235 "info=true/false\nWith the option \"info=true\" some of the steps involved in the evaluation of \+ the limit will be shown in a case where the limit is one of the standa rd types given at the early stages in a course on differential calculu s.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To make the proced ure active, open the subsection, place the cursor anywhere after the p rompt [ > and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "eval_lim/Limit: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44938 "eval_lim := proc(L)\n local fx,eq,x,a,prntflg,Options,fx2 ,val,nfx,facnfx,dfx,facdfx,cfx,cnfx,cdfx,\n left_lim,right_lim ,ngx,gx,ga,nn,n,m,dgx,rx,nrx,drx,fnrx,fdrx,fnmx,fdmx,\n tx,ff, v,i,j,lx,dx,qx,sx,tj,pw,fpx,flx,dd,ux,vx,phi,dir,delta,srx,pm,ax,\n \+ bx,r,px,dmx,nmx,znfx,rnfx,zdfx,rdfx,kx,flrs,b,hx,c,dux,d2ux,sgn, k,trms,\n stp,la,eax,ebx,er,t1,t2,simp,lsimp,asimp,d,e,wx,flg, drv,lft,rght,dvx,\n d2vx,dwx,d2wx,pws,mx,nr,dr,recip,rt,valid, nhx;\n\n if type(L,'function') and op(0,L)='Limit' and member(nops(L ),\{2,3\}) \n and type(op(1,L),algebraic) and type(op(2,L),name=a nything) then\n fx := op(1,L);\n eq := op(2,L);\n if no ps(L)=3 then \n dir := op(3,L);\n if not member(dir,\{ 'left','right'\}) then \n error \"invalid directional argum ent for the limit\"\n end if;\n else dir := NULL end if; \n x := op(1,eq);\n a := op(2,eq);\n if not type(a,alge braic) or has(a,x) then\n error \"invalid limiting point for l imit\"\n end if;\n else\n error \"the 1st argument, %1, is invalid .. it should be a limit described using 'Limit'\",L;\n end \+ if;\n\n # Get the options.\n # Set the default values to start wit h.\n prntflg := false;\n Options := [args[2..nargs]];\n if nargs >=2 then\n if not type(Options,list(equation)) then\n err or \"each optional argument must be an equation\"\n end if;\n \+ if hasoption(Options,'info','prntflg') then\n if prntflg<>tr ue then prntflg := false end if;\n end if;\n end if;\n\n ## h andle limits at infinity\n if abs(a)=infinity then\n if (a>0 an d dir='right') or (a<0 and dir=left) then\n error \"limit dire ction is inconsistent with infinite limiting point\"\n end if;\n \+ if prntflg then print(L) end if;\n ## some radicals can be s implified \"at infinity\"\n pws := [op(select(type,indets(fx),pol ynom(anything,x)^And(rational,Not(integer))))];\n if pws<>[] then \n gx := fx;\n hx := fx;\n kx := fx;\n \+ valid := true;\n for i to nops(pws) do\n ux := pws [i];\n vx := op(1,ux);\n r := op(2,ux);\n \+ n := degree(vx,x);\n c := coeff(vx,x,n); ## leading co efficient\n sgn := signum(0,c,0);\n m := r*n;\n \+ nr := numer(r);\n dr := denom(r);\n i f irem(dr,2)=0 then ## even root\n if (irem(n,2)=1 and a *sgn>0) or (irem(n,2)=0 and sgn>0) then\n if sgn>0 th en\n mx := c^r*x^m\n else ## avoi d mx := (-c)^r*(-x)^m \n fx := eval(fx,x=-x);\n \+ a := -a;\n if prntflg then\n \+ print(``);\n print(`The lim it has the same value as:`);\n end if;\n \+ return eval_lim(Limit(fx,x=a),info=prntflg);\n \+ end if;\n if a<0 and type(m,integer) and abs(ire m(m,2))=1 then mx := -mx end if;\n else ## avoid complex nos.\n if prntflg then\n print(f x,` is not defined as x approaches`,a); \n print(` so the limit does not exist.`);\n end if;\n \+ return `undefined`;\n end if;\n else \+ ## odd root\n if 0fx then\n if prntflg then print(``=Limit(gx,eq)) end if;\n fx := gx;\n if valid and denom(fx)<>1 then\n \+ d := -1;\n nhx := numer(hx);\n \+ if type(nhx,polynom(anything,x)) then\n d := degree( nhx); \n else ## a>0 then\n kx := expan d(nhx);\n if type(kx,`+`) then ##determine \"psuedo\" degree\n for i to nops(kx) do\n \+ if has(op(i,kx),x) then\n pm := patmat ch(op(i,kx),(aa::freeof(x))*x^(rt::rational),'la');\n \+ if pm then\n r :=subs(la,rt);\n end if;\n else\n \+ pm := true;\n r := 0; \n end if;\n if pm then \n if r>d then d := r end if;\n \+ else\n break;\n \+ end if;\n end do;\n end if; \n end if;\n if d>0 then\n \+ gx := expand(numer(fx)/x^d)/expand(denom(fx)/x^d);\n \+ if prntflg then print(``=Limit(gx,eq)) end if;\n fx := gx;\n end if;\n end if;\n val \+ := traperror(simplify(eval(gx,eq)));\n if val<>lasterror an d not has(val,`undefined`) then\n if prntflg then print( ``=val) end if;\n return val;\n end if;\n \+ end if; \n end if;\n \n if type(fx,ratpoly(anything,x )) then\n if type(fx,polynom) then\n d := degree(fx ,x);\n b := 0;\n px := expand(fx);\n \+ if prntflg and px<>fx then\n print(``=Limit(px,eq))\n \+ end if;\n else\n gx := normal(fx);\n \+ if type(gx,polynom) then\n if prntflg then\n \+ print(``=Limit(gx,eq))\n end if;\n \+ d := degree(gx,x);\n b := 0;\n px := \+ gx;\n else \n nfx := nu mer(gx);\n dfx := denom(gx);\n gx := expan d(nfx)/expand(dfx);\n if prntflg then\n \+ if gx<>fx then\n print(``=Limit(gx,eq));\n \+ end if;\n end if;\n c := degree (nfx,x); \n b := degree(dfx,x);\n d := c-b ; ## difference in degrees\n if d>=0 then e := b else e \+ := c end if;\n nfx := expand(nfx/x^e);\n d fx := expand(dfx/x^e);\n if prntflg then print(``=Limit( nfx/dfx,eq)) end if;\n if d>=0 then\n \+ px := expand(select(_U->degree(_U,x)>=0,nfx)/coeff(dfx,x,degree(dfx, x)))\n else\n px := coeff(nfx,x,degree( nfx,x))/select(_U->degree(_U,x)>=0,dfx)\n end if;\n \+ end if;\n end if; \n if d=0 then\n v al := px;\n if prntflg then print(``=val) end if;\n \+ return val;\n end if;\n if d>0 then\n s gn := signum(0,coeff(px,x,d),0); ## sign of leading coeff\n \+ if irem(d,2)=1 then\n val := sgn*a;\n if \+ prntflg then\n if sgn>0 then\n pr int(`The polynomial`,px,` has odd degree and the leading coefficient i s positive.`);\n else\n print(`Th e polynomial`,px,` has odd degree and the leading coefficient is negat ive.`);\n end if;\n end if;\n \+ else\n val := sgn*infinity;\n if prntflg then\n if sgn>0 then\n print(`Th e polynomial`,px,` has even degree and the leading coefficient is posi tive.`);\n else\n print(`The poly nomial`,px,` has even degree and the leading coefficient is negative.` );\n end if;\n end if;\n end if;\n if prntflg then print(`Hence`,L=val) end if;\n \+ else\n val := 0;\n if prntflg then print(``=v al) end if;\n end if;\n return val; \+ \n end if;\n\n ## some limits at infinity can be evaluated \+ by rationalizing\n if hastype(fx,`^`) then\n gx := fronte nd(factor,[fx]);\n if prntflg and gx<>fx then print(``=Limit(g x,eq)) end if;\n for j to 6 do\n if type(gx,`*`) th en\n mx,kx := selectremove(_U->`if`( numer(_U)=1,evalb(e val(1/_U,x=a)=0),\n evalb(eval(_U,x=a)=0)), gx)\n else\n v := `if`(numer(gx)=1,eval(1/gx, x=a),eval(gx,x=a));\n if v=0 or has(v,`undefined`) then \+ \n mx,kx := gx,1;\n else\n \+ mx,kx := 1,gx; \n end if;\n end if;\n \+ if mx=1 then break end if; \n if type(mx,`*`) \+ then \n hx := op(1,mx);\n \+ kx := kx*mx/hx;\n else\n hx := mx; \n \+ end if; \n recip := evalb(numer(hx)=1);\n \+ if recip then hx := 1/hx end if;\n if type(hx,`+`) and has type(fx,`^`) then\n nn := nops(hx);\n stp \+ := false;\n pm := false;\n for k from nn t o 1 by -1 do\n ## try grouping terms in different way s\n trms := combinat[choose](\{op(hx)\},k):\n \+ for i to nops(trms) do \n gx := add(trms [i,j],j=1..nops(trms[i]));\n gx := frontend(factor ,[gx]);\n pm := patmatch(gx,(ax::algebraic)*\n \+ (bx::And(ratpoly(anything,x),Not(freeof(x))))^\n \+ (rt::And(rational,Not(integer))),'la');\n \+ if pm then\n eax := expand(subs (la,ax));\n ebx := subs(la,bx);\n \+ if type(eax,ratpoly(anything,x)) and has(ebx,x) then\n \+ er := subs(la,rt);\n n \+ := denom(er);\n gx := eax*ebx^er;\n \+ ga := simplify(hx-gx); \n \+ if ga<>0 then\n stp := true;\n \+ break;\n end if;\n \+ end if;\n end if;\n \+ end do;\n if stp then break end if;\n \+ end do;\n hx := gx+ga;\n ga := -ga;\n \+ if pm and n>1 and type(gx^n,ratpoly(anything,x)) then \n \+ m := n-1;\n if recip then\n \+ qx := add(gx^(m-i)*ga^i,i=0..m);\n drx := expand(hx*qx);\n rx := qx/drx;\n \+ if prntflg then\n print(``=Limit((kx*``( qx))/(hx*qx),eq));\n if not (type(simplify(expa nd(denom(rx))/drx),integer) or \n type(simpl ify(drx/expand(denom(rx))),integer)) then \n \+ ## if denominator changes due to automatic cancellation\n \+ print(``=Limit(kx*qx/``(drx),eq));\n \+ end if;\n end if;\n else\n qx := add(gx^(m-i)*ga^i,i=0..m);\n \+ nrx := expand(hx*qx);\n rx := nrx/qx;\n \+ if prntflg then\n print(``=Limi t((hx*``(qx))/qx*kx,eq));\n if not (type(simpli fy(expand(numer(rx))/nrx),integer) or \n typ e(simplify(nrx/expand(numer(rx))),integer)) then \n \+ ## if numerator changes due to automatic cancellation\n \+ print(``=Limit(``(nrx)/qx*kx,eq));\n \+ end if;\n end if;\n e nd if;\n gx := rx*kx;\n if prntflg t hen print(``=Limit(gx,eq)) end if;\n end if;\n \+ end if;\n end do;\n val := traperror(simplify(eval(g x,eq)));\n if val<>lasterror and not has(val,`undefined`) then \n if prntflg then print(``=val) end if;\n retur n val;\n end if;\n end if;\n val := value(L);\n \+ if prntflg then print(``=val) end if;\n return val;\n end if; \+ ## limits at infinity\n\n ## Replace occurrences of sqrt(u^n),n even , by abs(u^(n/2)).\n asimp := false;\n if hastype(fx,'sqrt') then \n gx := fx;\n trms := [op(select(_U->(nops(_U)=2 and op(0,_ U)=`^`) and \n abs(op(2,_U))=1/2,indets(gx)))];\n for \+ i to nops(trms) do\n if patmatch(op(i,trms),\n condi tional(sqrt((ax::algebraic)^(k::integer)),irem(k,2)=0),'la') then\n \+ t1 := abs(subs(la,ax^(k/2)));\n gx := subs(op(i,tr ms)=t1,gx);\n trms := subs(op(i,trms)=t1,trms);\n e lif patmatch(op(i,trms),\n conditional(1/sqrt(ax::algebraic^ (k::integer)),irem(k,2)=0),'la') then\n t2 := abs(subs(la,a x^(k/2)));\n gx := subs(op(i,trms)=1/t2,gx);\n t rms := subs(op(i,trms)=1/t2,trms);\n end if;\n end do;\n \+ if gx<>fx then\n asimp := true;\n if prntflg then print(L=Limit(gx,eq,dir)) end if;\n fx := gx;\n end if; \n end if; \n\n ## Handle some functions involving 'floor' by simp lifying locally.\n lsimp := false;\n if has(fx,'floor') then\n \+ if has(fx,\{Heaviside,abs,signum,piecewise,min,max\}) then\n \+ return value(L)\n end if; \n gx := fx;\n flrs := [op (select(_U->op(0,_U)=floor,indets(gx)))];\n nn := nops(flrs);\n \+ for i to nn do\n ux := op(1,op(i,flrs));\n if has( ux,'floor') then ## move to end of list and try again\n flr s := [op(1..i-1,flrs),op(i+1..nn,flrs),op(i,flrs)];\n i := \+ i-1;\n next;\n end if;\n ux := simplify(ux) ;\n flg := 0;\n if op(0,ux)='piecewise' then\n \+ if nops(ux)=3 and (op(1,ux)=(x1 and flg<4 then\n c := traperror(f loor(simplify(limit(vx,x=a,'left'))));\n if c=lasterror \+ then return value(L) end if;\n d := traperror(floor(simp lify(limit(wx,x=a,'right'))));\n if c=lasterror then ret urn value(L) end if;\n if flg=2 then\n \+ t1 := piecewise(x0 then return value( L) end if;\n if type(b,integer) then\n drv := 0;\n while drv=0 do\n dux := diff(ux,x );\n drv := traperror(simplify(eval(dux,x=a)));\n \+ if drv=lasterror then return value(L) end if;\n \+ sgn := signum(0,drv,0);\n if sgn>0 then\n \+ ## ux is an increasing argument for floor\n \+ if flg=0 then\n t1 := piecewise(x0 then\n \+ ## minimum for argument of floor\n d := fl oor(b);\n if flg=0 then\n \+ t1 := d\n else\n \+ t1 := piecewise(x=a,e,d)\n end if;\n \+ gx := subs(op(i,flrs)=t1,gx);\n f lrs := subs(op(i,flrs)=t1,flrs);\n elif sgn<0 then \n ## maximum for argument of floor\n \+ d := floor(b);\n if flg=0 then\n \+ t1 := d-1\n else\n \+ t1 := piecewise(x=a,e,d-1)\n \+ end if;\n gx := subs(op(i,flrs)=t1,gx);\n flrs := subs(op(i,flrs)=t1,flrs);\n \+ end if;\n end if;\n ux := \+ d2ux;\n end do; \n else ## b is not an integer\n d := floor(b);\n if f lg=0 then\n lsimp := true;\n t1 := d ;\n else\n t1 := piecewise(x=a,e,d)\n \+ end if;\n gx := subs(op(i,flrs)=t1,gx);\n \+ flrs := subs(op(i,flrs)=t1,flrs);\n end if;\n \+ end if;\n if flg>3 then\n ## Check for cont inuity of vx at x=a\n b := traperror(simplify(eval(vx,x=a)) );\n if b=lasterror then return value(L) end if;\n \+ d := traperror(simplify(limit(vx,x=a)));\n if d=lasterro r or signum(0,b-d,0)<>0 then return value(L) end if;\n if t ype(b,integer) then\n ## if vx is increasing floor value is b-1 \n drv := 0;\n lft := b;\n \+ while drv=0 do\n dvx := diff(vx,x);\n \+ drv := traperror(simplify(eval(dvx,x=a)));\n \+ if drv=lasterror then return value(L) end if;\n sgn \+ := signum(0,drv,0);\n if sgn>0 then lft := lft-1\n \+ elif sgn<0 then break\n else # sgn=0\n d2vx := diff(dvx,x); ## 2nd deriv,4th deriv ...\n drv := traperror(simplify(eval(d2vx,x=a)));\n \+ if drv=lasterror then return value(L) end if;\n \+ sgn := signum(0,drv,0);\n if sgn<0 \+ then lft := lft-1\n elif sgn>0 then break\n \+ end if;\n end if;\n vx := d2vx;\n end do; \n else \+ ## b is not an integer\n lft := floor(b);\n e nd if;\n ## Check for continuity of wx at x=a\n \+ b := traperror(simplify(eval(wx,x=a)));\n if b=lasterror th en return value(L) end if;\n d := traperror(simplify(limit( wx,x=a)));\n if d=lasterror or signum(0,b-d,0)<>0 then retu rn value(L) end if;\n if type(b,integer) then\n \+ ## if wx is decreasing floor value is b-1 \n drv := 0 ;\n rght := b;\n while drv=0 do\n \+ dwx := diff(wx,x);\n drv := traperror(simpli fy(eval(dwx,x=a)));\n if drv=lasterror then return va lue(L) end if;\n sgn := signum(0,drv,0);\n \+ if sgn<0 then rght := rght-1\n elif sgn>0 then break\n else # sgn=0 \n d2wx := \+ diff(dwx,x); ## 2nd deriv,4th deriv ...\n drv := t raperror(simplify(eval(d2wx,x=a)));\n if drv=laste rror then return value(L) end if;\n sgn := signum( 0,drv,0);\n if sgn<0 then rght := rght-1\n \+ elif sgn>0 then break\n end if;\n \+ end if;\n wx := d2wx;\n e nd do; \n else ## b is not an integer\n \+ rght := floor(b);\n end if;\n if flg =4 then\n t1 := piecewise(xfx then print(`Then`,'phi'(x)=fpx) end if;\n end if;\n ## modify rational expressions to avoid unwanted simplification \+ during conversion\n fpx := map(_U->`if`(type(_U,And(ratpoly(anyth ing,x),Not(polynom(anything,x)))),\n ``(numer(_U))/``(denom(_ U)),_U),fpx); \n flx := traperror(eval(subs(``=(_U->_U),conv ert(fpx,'pwlist',x))));\n if flx=lasterror then\n ## may \+ have a specified value at x=a which can be ignored\n t1 := [op (fpx)];\n if member(eq,t1) then\n nn := nops(t1);\n for i to nn do\n if op(i,t1)=eq then\n \+ v := op(i+1,t1);\n t2 := [op(1..i-1,t1),o p(i+2..nn,t1)];\n simp := true;\n br eak;\n end if;\n end do;\n if simp then\n if prntflg then\n print(`The va lue of the limit is not affected by the value`,v,` at`,eq);\n \+ print(``);\n end if;\n if nops(t2 )=1 then\n if type(op(t2),And(realcons,Not(infinity)) ) then\n val := op(t2);\n if p rntflg then\n print(`Hence`,Limit('phi'(x),eq,d ir)=val);\n end if;\n return v al;\n else\n fx := eval(subs(``=( _U->_U),op(t2)));\n goto(1111);\n \+ end if; \n else\n fpx := pie cewise(op(t2));\n flx := traperror(eval(subs(``=(_U-> _U),convert(fpx,'pwlist',x))));\n if flx=lasterror th en\n error \"could not convert piecewise function \+ to a piecewise list\"\n end if;\n end i f;\n end if;\n end if;\n end if;\n if has (flx,piecewise) then\n error \"could not convert piecewise fun ction to a piecewise list\"\n end if;\n dd := [seq(flx[2*i], i=1..nops(flx)/2)];\n if member(a,dd) then\n for i to nop s(dd) do\n if signum(0,dd[i]-a,0)=0 then break end if;\n \+ end do;\n ux := flx[2*i-1];\n vx := flx[2*i+1]; \n if dir=NULL then\n if prntflg then\n \+ print(Limit('phi'(x),eq,'left')=Limit(ux,eq,'left'));\n \+ print(``);\n left_lim := eval_lim(Limit(ux,eq,'left') ,info=true);\n if left_lim='undefined' then\n \+ print(`Hence`,Limit('phi'(x),eq,'left'),` does not exist.`)\n \+ else\n print(`Hence`,Limit('phi'(x),eq, 'left')=left_lim);\n end if;\n print(``); \n print(Limit('phi'(x),eq,'right')=Limit(vx,eq,'right') );\n right_lim := eval_lim(Limit(vx,eq,'right'),info=tru e);\n if right_lim='undefined' then\n p rint(`Hence`,Limit('phi'(x),eq,'right'),` does not exist.`)\n \+ else\n print(`Hence`,Limit('phi'(x),eq,'right' )=right_lim);\n end if;\n print(``);\n \+ if left_lim=right_lim then\n print(Limit(' phi'(x),eq,'left')=Limit('phi'(x),eq,'right'));\n pri nt(`so`,Limit('phi'(x),eq)=left_lim);\n print(``);\n \+ else\n print(Limit('phi'(x),eq,'left')< >Limit('phi'(x),eq,'right'));\n print(`so`,Limit('phi '(x),eq),`does not exist.`);\n print(``);\n \+ end if;\n else \n left_lim := limit(ux,e q,'left');\n right_lim := limit(vx,eq,'right');\n \+ end if;\n if left_lim=right_lim then return left_lim e lse return `undefined` end if;\n elif dir='left' then\n \+ if prntflg then\n print(Limit('phi'(x),eq,'left')=L imit(ux,eq,'left'));\n print(``);\n left_l im := eval_lim(Limit(ux,eq,'left'),info=true);\n if left _lim='undefined' then\n print(`Hence`,Limit('phi'(x), eq,'left'),` does not exist.`)\n else\n \+ print(`Hence`,Limit('phi'(x),eq,'left')=left_lim);\n e nd if;\n print(``);\n else\n le ft_lim := limit(ux,eq,'left');\n end if;\n retur n left_lim; \n else ## dir='right'\n if prntflg the n\n print(Limit('phi'(x),eq,'right')=Limit(vx,eq,'right' ));\n print(``);\n right_lim := eval_lim(L imit(vx,eq,'right'),info=true);\n if right_lim='undefine d' then\n print(`Hence`,Limit('phi'(x),eq,'right'),` \+ does not exist.`)\n else\n print(`Henc e`,Limit('phi'(x),eq,'right')=right_lim);\n end if;\n \+ else\n right_lim := limit(ux,eq,'right');\n \+ end if;\n return right_lim;\n end if; \+ \n else\n dd := [-infinity,op(dd),infinity];\n f or i to nops(dd) do\n if signum(0,dd[i]-a,0)=-1 and signum( 0,dd[i+1]-a,0)=1 then break end if;\n end do;\n vx := \+ flx[2*i-1];\n if prntflg then\n print(Limit('phi'(x ),eq,dir)=Limit(vx,eq,dir));\n val := eval_lim(Limit(vx,eq, dir),info=true);\n if val='undefined' then\n \+ print(`Hence`,Limit('phi'(x),eq,dir),` does not exist.`)\n \+ else\n print(`Hence`,Limit('phi'(x),eq,dir)=val);\n \+ end if;\n else\n val := limit(vx,eq,dir); \+ \n end if;\n return val;\n end if;\n \+ end if;\n 1111:\n\n ## depart from Maple behaviour for functions i nvolving square roots etc.\n if type(a,And(realcons,Not(infinity))) \+ and hastype(fx,`^`) then\n if dir='left' then\n srx := se ries(eval(fx,x=a-delta),delta,6);\n if has(srx,Complex(1)) the n ## (real) left limit is not defined\n if prntflg then\n \+ print(fx,cat(` is not defined as `,x,` approaches`),a,` f rom the left,`);\n print(`so`,Limit(fx,eq,'left'),` does not exist.`);\n print(``);\n end if;\n \+ return `undefined`;\n end if;\n elif dir='right' th en \n srx := series(eval(fx,x=a+delta),delta,6);\n if \+ has(srx,Complex(1)) then ## (real) right limit is not defined\n \+ if prntflg then\n print(fx,cat(` is not defined as \+ `,x,` approaches`),a,` from the right,`);\n print(`so`,L imit(fx,eq,'right'),` does not exist.`);\n print(``);\n \+ end if;\n return `undefined`;\n end if; \n else ## two-sided limit\n srx := series(eval(fx,x=a-de lta),delta,6);\n if has(srx,Complex(1)) then ## (real) left li mit is not defined\n if prntflg then\n print( fx,cat(` is not defined as `,x,` approaches`),a,` from the left,`);\n \+ print(`so`,Limit(fx,eq,'left'),` does not exist.`);\n \+ print(`Hence`,L,` does not exist.`);\n print (``);\n end if;\n return `undefined`;\n \+ end if;\n srx := series(eval(fx,x=a+delta),delta,6);\n \+ if has(srx,Complex(1)) then ## (real) right limit is not defined\n \+ if prntflg then\n print(fx,cat(` is not defined as `,x,` approaches`),a,` from the right,`);\n print(`s o`,Limit(fx,eq,'right'),` does not exist.`);\n print(`He nce`,L,` does not exist.`);\n print(``);\n en d if;\n return `undefined`;\n end if;\n end if ;\n end if; \n\n if prntflg then\n if simp then print(`H ence`,Limit('phi'(x),eq,dir)=Limit(fx,eq,dir))\n elif lsimp then \+ print(`Hence`,L=Limit(fx,eq,dir))\n elif not asimp then print(L) \+ end if;\n end if;\n\n if type(fx,`+`) and not type(fx,polynom(anyt hing,x)) then\n lx := [op(fx)]; ## form list of terms \n if \+ type(lx,list(ratpoly(anything,x))) then\n dx := lcm(op(map(den om,[op(lx)]))); ## common denominator\n rx := add(expand(numer (lx[i])*quo(dx,denom(lx[i]),x)),i=1..nops(lx))/dx;\n if prntfl g then\n sx := 0;\n for j to nops(lx) do\n \+ tj := expand(numer(lx[j])*quo(dx,denom(lx[j]),x));\n \+ if type(tj,`+`) then tj := ``(tj) end if;\n sx := \+ sx + tj;\n end do;\n qx := sx/dx;\n i f qx<>rx and qx<>fx then print(``=Limit(qx,eq,dir)) end if;\n \+ if denom(rx)<>dx then\n sx := ``(eval(subs(``=(_U->_U ),sx))); \n print(``=Limit(sx/dx,eq,dir));\n \+ end if;\n print(``=Limit(rx,eq,dir));\n end if;\n \+ fx := rx;\n end if;\n end if;\n\n if type(fx,`+`) then \n nfx := numer(fx);\n dfx := denom(fx);\n if simplify( eval(nfx,eq))=0 and simplify(eval(dfx,eq))=0 then \n rx := nfx /dfx;\n if rx<>fx then\n if prntflg then\n \+ print(``=Limit(rx,eq,dir));\n end if;\n fx := rx;\n end if;\n end if;\n end if;\n\n if type(fx, polynom(anything,x)) then\n val := simplify(eval(fx,eq));\n \+ if prntflg then\n print(``=val);\n end if;\n return \+ val;\n elif type(fx,`+`) then\n n := nops(fx);\n if prntfl g then\n print(``=add(Limit(op(j,fx),eq,dir),j=1..n));\n \+ print(``);\n end if;\n v := table(); \n for i to n d o\n v[i] := eval_lim(Limit(op(i,fx),eq,dir),info=prntflg);\n \+ end do;\n val := add(v[j],j=1..n);\n if prntflg then\n \+ print(``);\n print(L);\n print(``=add(Limit(op( j,fx),eq,dir),j=1..n));\n print(``=val);\n end if;\n \+ return val;\n elif type(fx,ratpoly(anything,x)) then\n nfx := \+ numer(fx);\n dfx := denom(fx);\n if simplify(eval(nfx,eq))<> 0 or simplify(eval(dfx,eq))<>0 then\n val := limit(fx,eq,dir); \n if val<>'undefined' then \n if prntflg th en print(``=val) end if; \n else # must be two-sided limi t with unequal infinite left and right limits\n left_lim := limit(fx,eq,'left');\n right_lim := limit(fx,eq,'right'); \n print(Limit(fx,eq,'left')=left_lim,` while `,Limit(fx ,eq,'right')=right_lim);\n print(`so the (two-sided) limit \+ does not exist.`);\n end if;\n return val;\n end \+ if;\n if prntflg then\n if type(fx,`*`) and nops(fx)=2 an d type(normal(op(1,fx)),`*`) then\n print(``=Limit(``(norma l(op(1,fx)))*op(2,fx),eq,dir));\n end if; \n end if; \+ \n cfx := gcd(nfx,dfx,'cnfx','cdfx');\n if cfx=1 then error \"unexplained error\" end if;\n if degree(nfx)<=degree(dfx) then \n nfx := ``(cfx)*cnfx;\n dfx := cfx*cdfx;\n else \n nfx := cfx*cnfx;\n dfx := ``(cfx)*cdfx;\n end \+ if; \n if prntflg then\n print(``=Limit(nfx/dfx,eq,d ir));\n print(``=Limit(cnfx/cdfx,eq,dir));\n end if;\n \+ val := limit(fx,eq,dir);\n if val<>'undefined' then \n \+ if prntflg then print(``=val) end if; \n else # must be two-sided limit with unequal infinite left and right limits\n \+ left_lim := limit(fx,eq,'left');\n right_lim := limit(fx,e q,'right');\n print(Limit(fx,eq,'left')=left_lim,` while `, Limit(fx,eq,'right')=right_lim);\n print(`so the (two-sided) l imit does not exist.`);\n end if;\n return val;\n else ## \+ handle some other limits of type 0/0 involving roots\n if type(fx ,`*`) and nops(fx)=2 and type(normal(op(1,fx)),`*`) then\n ux \+ := ``(normal(op(1,fx)))*op(2,fx);\n if prntflg then print(``=L imit(ux,eq,dir)) end if;\n vx := normal(eval(subs(``=(_U->_U), ux)));\n if vx<>fx then\n fx := vx;\n if prntflg then print(``=Limit(vx,eq,dir)) end if; \n end if;\n \+ end if;\n nfx := numer(fx);\n dfx := denom(fx);\n \+ if simplify(eval(nfx,eq))=0 and simplify(eval(dfx,eq))=0 then\n \+ if type(nfx,`*`) then\n znfx,rnfx := selectremove(_U->eva lb(simplify(eval(_U,eq)=0)),nfx);\n else\n znfx,rnf x := nfx,1;\n end if;\n if type(dfx,`*`) then\n \+ zdfx,rdfx := selectremove(_U->evalb(simplify(eval(_U,eq)=0)),dfx) ;\n else\n zdfx,rdfx := dfx,1;\n end if;\n \+ kx := rnfx/rdfx;\n if kx<>1 then\n nfx := z nfx;\n dfx := zdfx;\n end if;\n if type(nfx ,`+`) and not type(nfx,polynom(anything,x)) then\n nn := no ps(nfx);\n stp := false;\n pm := false;\n \+ for k from nn to 1 by -1 do\n ## try grouping terms in different ways\n trms := combinat[choose](\{op(nfx) \},k):\n for i to nops(trms) do \n gx : = add(trms[i,j],j=1..nops(trms[i]));\n gx := frontend (factor,[gx]);\n pm := patmatch(gx,(ax::algebraic)*\n (bx::And(polynom(anything,x),Not(freeof(x))))^\n (r::And(rational,Not(integer))),'la');\n \+ if pm then\n eax := expand(subs(la,ax)); \n ebx := subs(la,bx);\n if ty pe(eax,polynom(anything,x)) and has(ebx,x) then\n \+ er := subs(la,r);\n n := denom(er);\n \+ gx := eax*ebx^er;\n ga := simp lify(nfx-gx); \n if ga<>0 then\n \+ stp := true;\n break;\n \+ end if;\n end if;\n \+ end if;\n end do;\n if stp then break en d if;\n end do;\n nfx := gx+ga;\n ga \+ := -ga;\n if pm and n>1 and type(gx^n,polynom(anything,x)) \+ then \n m := n-1;\n ngx := add(gx^(m-i)*ga ^i,i=0..m);\n nrx := expand(nfx*ngx)*numer(kx);\n \+ rx := nrx/(dfx*ngx*denom(kx));\n if prntflg then \n print(``=Limit((nfx*``(ngx))/(dfx*ngx)*kx,eq,dir)) ;\n if not (type(simplify(expand(numer(rx))/nrx),inte ger) or \n type(simplify(nrx/expand(numer(rx))),in teger)) then \n ## if numerator changes due to a utomatic cancellation\n print(``=Limit(``(expand(n fx*ngx))/(dfx*ngx)*kx,eq,dir));\n end if;\n \+ print(``=Limit(rx,eq,dir));\n end if;\n \+ nrx := numer(rx);\n drx := denom(rx);\n \+ ## look for further cancellation\n fnrx := frontend( factor,[nrx]);\n fdrx := frontend(factor,[drx]); \+ \n ## determine common factors of numerator and denominator\n fnmx := `if`(type(fnrx,`*`),[op(fnrx)],[f nrx]);\n fdmx := `if`(type(fdrx,`*`),[op(fdrx)],[fdrx]); \n ff := select(_U->member(_U,fnmx) or member(-_U,fnmx), fdmx);\n tx := mul(ff[i],i=1..nops(ff));\n \+ if tx<>1 then\n ux := ``(tx)*normal(fnrx/tx)/fdrx; \+ \n if prntflg then print(``=Limit(ux,eq,dir)) end if; \n vx := eval(subs(``=(_U->_U),ux));\n \+ else\n vx := fnrx/fdrx;\n end if;\n \+ if vx<>rx then\n if prntflg then print(``= Limit(vx,eq,dir)) end if;\n rx := vx; \n \+ end if;\n ux := simplify(rx);\n if ux< >rx then\n if prntflg then print(``=Limit(ux,eq,dir)) end if;\n rx := ux; \n end if;\n \+ val := traperror(simplify(eval(numer(rx),eq))/simplify(eval (denom(rx),eq)));\n if val<>lasterror then\n \+ val := simplify(val);\n if prntflg then print(` `=val) end if;\n return val;\n else\n \+ if simplify(eval(nrx,eq))=0 and simplify(eval(drx,eq)) =0 then\n fx := rx;\n nfx := n umer(fx);\n dfx := denom(fx);\n \+ if type(nfx,`*`) then\n znfx,rnfx := selectre move(_U->evalb(simplify(eval(_U,eq)=0)),nfx);\n el se\n znfx,rnfx := nfx,1;\n \+ end if;\n if type(dfx,`*`) then\n \+ zdfx,rdfx := selectremove(_U->evalb(simplify(eval(_U,eq)=0)),df x);\n else\n zdfx,rdfx := d fx,1;\n end if;\n kx := rnfx/r dfx; \n if kx<>1 then\n nfx := znfx;\n dfx := zdfx;\n \+ end if;\n end if;\n end if;\n \+ end if;\n end if; \n if type(dfx,`+`) and not type( dfx,polynom(anything,x)) then\n nn := nops(dfx);\n \+ stp := false;\n pm := false;\n for k from nn \+ to 1 by -1 do\n ## try grouping terms in different ways \n trms := combinat[choose](\{op(dfx)\},k):\n \+ for i to nops(trms) do\n gx := add(trms[i,j],j=1. .nops(trms[i]));\n gx := frontend(factor,[gx]);\n \+ pm := patmatch(gx,(ax::algebraic)*\n \+ (bx::And(polynom(anything,x),Not(freeof(x))))^\n \+ (r::And(rational,Not(integer))),'la');\n if pm then \n eax := expand(subs(la,ax));\n \+ ebx := subs(la,bx);\n if type(eax,polynom(anyth ing,x)) and has(ebx,x) then\n er := subs(la,r); \n n := denom(er);\n gx \+ := eax*ebx^er;\n ga := simplify(dfx-gx); \n \+ if ga<>0 then\n stp := t rue;\n break;\n end i f;\n end if;\n end if;\n \+ end do;\n if stp then break end if;\n e nd do;\n dfx := gx+ga;\n ga := -ga;\n \+ if pm and n>1 and type(gx^n,polynom(anything,x)) then\n \+ m := n-1;\n dgx := add(gx^(m-i)*ga^i,i=0..m);\n \+ drx := expand(dfx*dgx)*denom(kx);\n rx := (nfx*dg x*numer(kx))/drx;\n if prntflg then\n p rint(``=Limit((nfx*dgx)/(dfx*``(dgx))*kx,eq,dir));\n \+ if not (type(simplify(expand(denom(rx))/drx),integer) or \n \+ type(simplify(drx/expand(denom(rx))),integer)) then \n \+ ## if denominator changes due to automatic cancellation \n print(``=Limit((nfx*dgx)/``(expand(dfx*dgx))*kx ,eq,dir));\n end if;\n print(``=Limi t(rx,eq,dir));\n end if;\n nrx := numer(rx );\n drx := denom(rx);\n fnrx := frontend( factor,[nrx]);\n fdrx := frontend(factor,[drx]);\n \+ ## determine common factor of numerator and denominator\n \+ fnmx := `if`(type(fnrx,`*`),[op(fnrx)],[fnrx]);\n \+ fdmx := `if`(type(fdrx,`*`),[op(fdrx)],[fdrx]);\n f f := select(_U->member(_U,fnmx) or member(-_U,fnmx),fdmx);\n \+ tx := mul(ff[i],i=1..nops(ff));\n if tx<>1 then\n \+ ux := nrx/(``(tx)*normal(fdrx/tx));\n \+ if prntflg then print(``=Limit(ux,eq,dir)) end if;\n \+ vx := eval(subs(``=(_U->_U),ux));\n else\n \+ vx := fnrx/fdrx;\n end if; \n if vx<> rx then\n if prntflg then print(``=Limit(vx,eq,dir)) \+ end if;\n rx := vx; \n end if;\n \+ ux := simplify(rx);\n if ux<>rx then\n \+ if prntflg then print(``=Limit(ux,eq,dir)) end if;\n \+ rx := ux; \n end if;\n val := \+ traperror(simplify(eval(numer(rx),eq))/simplify(eval(denom(rx),eq))); \n if val<>lasterror then\n val := simp lify(val);\n if prntflg then print(``=val) end if;\n \+ return val;\n else\n f x := rx;\n end if;\n end if;\n end if ;\n val := limit(fx,eq,dir);\n if prntflg then print(` `=val) end if;\n return val;\n end if;\n if type(dfx ,polynom(anything,x)) and member(x,indets(dfx)) and \n simplify (eval(dfx,eq))=0 and simplify(eval(nfx,eq))<>0 then\n val := v alue(L); \n if prntflg and dir=NULL and val='undefined' then \n left_lim := limit(fx,eq,'left');\n right_lim \+ := limit(fx,eq,'right');\n print(Limit(fx,eq,'left')=left_l im,` while `,Limit(fx,eq,'right')=right_lim);\n print(`s o the (two-sided) limit does not exist.`);\n else\n \+ print(``=val);\n end if;\n return val;\n end if; \n val := limit(fx,eq,dir);\n if prntflg then\n ### \+ Can possibly scrap most of the following\n if val='undefined' \+ then\n if dir=NULL then\n left_lim := limit(f x,eq,'left');\n right_lim := limit(fx,eq,'right');\n \+ if left_lim='undefined' and right_lim='undefined' then\n \+ print(`The limit does not exist.`);\n eli f left_lim='undefined' and right_lim<>'undefined' then\n \+ print(Limit(fx,eq,'left'),`does not exist, while `,Limit(fx,eq,' right')=right_lim);\n print(`so the (two-sided) limit does not exist.`);\n elif right_lim='undefined' and lef t_lim<>'undefined' then\n print(Limit(fx,eq,'left')=l eft_lim,` while `,Limit(fx,eq,'right'),`does not exist,`);\n \+ print(`so the (two-sided) limit does not exist.`);\n \+ elif left_lim<>right_lim then\n print(Limit(f x,eq,'left')=left_lim,` while `,Limit(fx,eq,'right')=right_lim);\n \+ print(`so the (two-sided) limit does not exist.`);\n \+ end if;\n end if;\n ## one-sided l imit is undefined\n print(`The (one-sided) limit does not e xist.`);\n else \n print(``=val);\n end if; \n end if;\n return val; \n end if;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 86 ": The limits in the following examples have been found by using the special procedure " }{TEXT 0 8 "eval_lim" } {TEXT -1 60 " which can be loaded from the code in the previous sectio n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 " Given " }{XPPEDIT 18 0 "f(x)=PIECEWISE([x ^2+2,x<>1],[1,x=1])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&*$F'\"\"#\"\" \"F.F/0F'F/7$F//F'F/" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "Limit(f(x) ,x=1)" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"\"" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "f := x -> piecewise(x=1,1,x^2+2):\n'f(x)'=f(x);\nLimit(f(x),x=1): \neval_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#% \"xG-%*PIECEWISEG6$7$\"\"\"/F'F,7$,&*$)F'\"\"#F,F,F2F,%*otherwiseG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%$LetG/-%$phiG6#%\"xG-%*PIECEWISEG6$7$ \"\"\"/F(F-7$,&*$)F(\"\"#F-F-F3F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%TThe~value~of~the~limit~is~not~affected~by~the~valueG \"\"\"%$~atG/%\"xGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%&HenceG/-%&LimitG6$-%$phiG6#%\"xG/F+\"\"\"- F&6$,&*$)F+\"\"#F-F-F3F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\" $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 430 "p1 := plot(x^2+2,x=-2.5..2.5):\np2 := plot([[[1,3]]$ 2],style=point,symbol=[circle$2],symbolsize=[15,18],color=brown):\np3 \+ := plot([[[1,1]]$4],style=point,symbol=[circle$2,diamond,cross],color= red,\n symbolsize=[15,10$3]):\nt1 := plots[textplot]([[2.5 ,-.2,`x`],[-.15,6.5,`y`]],color=COLOR(RGB,.01,0,0)):\nplots[display]([ p||(1..3),t1],view=[-2.5..2.5,-.2..6.5],labels=[``,``],tickmarks=[4,3] ,\n title=\"graph of y = f(x)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 358 330 330 {PLOTDATA 2 "6/-%'CURVESG6$7S7$$!3++++++++D!#<$\"3+++++++] #)F*7$$!3smm;HU,\"R#F*$\"3Hpu2W!\\pr(F*7$$!3=L$3FH'='H#F*$\"3!)Q;#3\\r CF(F*7$$!3gmmTgBa*=#F*$\"3k()*e![d4%z'F*7$$!3wmm\"H_\">#3#F*$\"39Gg5Q: _NjF*7$$!3ML$3_!4Nv>F*$\"3+#Q2()>6?!fF*7$$!3km;/wfHw=F*$\"3%4:5(*e'[?b F*7$$!3;+]PM.tth9&F*7$$!3em;/,oln;F*$\"3;D011#z5y%F*7$$! 3%)**\\(oWB>c\"F*$\"3^!)y*Q&[gRWF*7$$!3;LL$epjJX\"F*$\"3\">*z)os%o6TF* 7$$!3amm\"z/otN\"F*$\"3*=50v,[C%QF*7$$!3))****\\P[_\\7F*$\"3?,%H&>BJhN F*7$$!3#*****\\7)Q79\"F*$\"345k:FgU-LF*7$$!3'*****\\i^)o.\"F*$\"3;^E@S 38vIF*7$$!3vlmT50A@%*!#=$\"3ac*e!fRf()GF*7$$!3OKLLeaR%H)F`p$\"3-1A>g*p zo#F*7$$!3kJLLLo#)RtF`p$\"3C+KVz0tQDF*7$$!3f***\\PfO%HiF`p$\"3%e^bF!)e !)Q#F*7$$!3/MLL$3`lC&F`p$\"3+XBc#>j_F#F*7$$!3q'**\\P4u\"oTF`p$\"3)f3eF vOP<#F*7$$!3*z**\\7G-89$F`p$\"3Ak=A+yn)4#F*7$$!3%)GL$3Fp)p?F`p$\"3/,M) zeVG/#F*7$$!3YKL3-$ff3\"F`p$\"3K8y0wIz6?F*7$$!3)Grmm\"z%zY#!#?$\"3E>pw !41++#F*7$$\"3!fL$e*)>px5F`p$\"33@TC+Uh6?F*7$$\"3w++v$f4t.#F`p$\"3-ey! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Limit((abs(x+1)-ab s(x-1))/x,x=0):\neval_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$LetG/-%$phiG6#%\"xG*&,&-%$absG6#,&F(\"\"\"F/F/F/-F,6#,&F(F/F/! \"\"F3F/F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%ThenG/-%$phiG6#%\"xG -%*PIECEWISEG6%7$,$*&\"\"#\"\"\"F(!\"\"F12F(F17$F/2F(F07$,$*&F/F0F(F1F 01F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%$phiG6#%\"xG/F *\"\"!-F%6$\"\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$\"\"# /%\"xG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%&HenceG/-%&LimitG6$-%$phiG6#%\"xG/F+\"\"!\"\"# " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := x -> (abs(x+1)-abs(x-1))/x:\n'f(x)'=f(x);\nplot( f(x),x=-7..7,y=0..2.2,tickmarks=[7,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&-%$absG6#,&F'\"\"\"F.F.F.-F+6#,&F'F.F.!\"\"F2F .F'F2" }}{PARA 13 "" 1 "" {GLPLOT2D 625 222 222 {PLOTDATA 2 "6&-%'CURV ESG6$7[p7$$!\"(\"\"!$\"3)p&G9dG9dG!#=7$$!3+nmmT)R[p'!#<$\"31S]TAaP()HF -7$$!3#HL$e>;KHkF1$\"3(o7q!)4[26$F-7$$!3Lmm;4'=28'F1$\"3oQcb6.EiKF-7$$ !3vmm;ki8IeF1$\"3,yMs_:XIMF-7$$!3+LLeMD)4`&F1$\"3')4TihS*fh$F-7$$!3)om ;HtGOD&F1$\"32-z/MD*o!QF-7$$!3S***\\i$\\Wm\\F1$\"3i?5$*=a-FSF-7$$!3Mmm \"H/R%pYF1$\"3g:'fvtpJG%F-7$$!3W++D^cQtVF1$\"3-eE!Q-;Jd%F-7$$!3[KLL[$e )oSF1$\"3)Q2h#*p$Q:\\F-7$$!3Tmm;M0j+QF1$\"3G`=gBZGi_F-7$$!3Q*****\\ap' )\\$F1$\"3*y!et6(ekr&F-7$$!3')*****\\noa>$F1$\"3G?M?%zi)eiF-7$$!3q**** *\\XyK!HF1$\"3/t()fkRw))oF-7$$!3]mm\"HuTzj#F1$\"3WP>Lp&o;e(F-7$$!3(HLL $G2VABF1$\"3l)Q%)G(pm6')F-7$$!3#GLL37\"z)=#F1$\"3*3bdz'RYP\"*F-7$$!3oK LL8::b?F1$\"3t%H=>fU;t*F-7$$!3Am;zzop**=F1$\"39*ow$R&*z_5F17$$!3!)*** \\iCUUu\"F1$\"3K9-!4sHm9\"F17$$!3im;za&Qmg\"F1$\"3$pF4*e]$[C\"F17$$!3W LLLj[.p9F1$\"3%)\\.Ux!Q9O\"F17$$!3))*\\iSL[NR\"F1$\"3BLS!)R_=N9F17$$!3 Nm;z/=1=8F1$\"3K*[%eG&zt^\"F17$$!3!G$3_v_dU7F1$\"34M_?M/c4;F17$$!3C*** \\iu)3n6F1$\"31I,,Sdm8x?&4\"F1$\"3sI`].w8E=F17$$!33**** \\#pEL-\"F1$\"3'=I7r25W&>F17$$!3+\\7yNkN05F1$\"3%>N;b?W$*)>F17$$!3O!* \\i!zhQ()*F-$\"3y**************>F17$$!3q!\\PMAfTp*F-$\"\"#F*7$$!3$**)* \\ilcW^*F-Fbu7$$!3]*)\\(=_^]:*F-$\"3W+++++++?F17$$!3/*)**\\(QYcz)F-Fbu 7$$!3cdm\"Hz>hROC)*F-Fbu7$ $\"38.v=U;0=**F-Fju7$$\"3o3xJKR<,5F1$\"3SF17$$\"3%pm;/qU0,\" F1$\"3a(opqdM\"z>F17$$\"3WL$3_%G\\&3\"F1$\"39#3tD0\"[U=F17$$\"3'****** **)HWg6F1$\"3!yp;\\vzMs\"F17$$\"3;+]P%3nPB\"F1$\"3a6*fR^^5i\"F17$$\"3O ++vy6428F1$\"3?Lg\\Q^6I:F17$$\"3c+]7t_T!Q\"F1$\"3G9O(QPR)[9F17$$\"3w++ ]n$RPX\"F1$\"35paoECwv8F17$$\"3%4+D1.Hcf\"F1$\"3E+X_qV(*F-7$$\"3mKLe%znT>#F1$\"3Rf8LsJ2:\"*F-7$$\"3olmmO$ GdL#F1$\"3%H9U7iREc)F-7$$\"3t,++D0-QEF1$\"3%))o'p\\@W\"e(F-7$$\"3qKL3x @%>\"HF1$\"3SF0w\\#o#ooF-7$$\"3*>++]*3T6KF1$\"3LDNH'G#zFiF-7$$\"3ImmT? w=$\\$F1$\"3H&4srlHas&F-7$$\"3[++v)[Dxy$F1$\"3Ye+P&p8-G&F-7$$\"3'pmm;4 !pvSF1$\"3=h4vYW92\\F-7$$\"3M***\\PMirP%F1$\"3_Z!G!f- " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x)=(x^3-x)/(x-1)" "6#/-%\"fG6#% \"xG*&,&*$F'\"\"$\"\"\"F'!\"\"F,,&F'F,F,F-F-" }{TEXT -1 12 ", find (a ) " }{XPPEDIT 18 0 "Limit(f(x),x = 1);" "6#-%&LimitG6$-%\"fG6#%\"xG/F) \"\"\"" }{TEXT -1 11 " and (b) " }{XPPEDIT 18 0 "Limit(f(x),x = -1); " "6#-%&LimitG6$-%\"fG6#%\"xG/F),$\"\"\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Limit((x^3-x)/(x-1),x=1):\neval_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&*$)%\"xG\"\"$\"\"\"F,F*!\"\"F,,& F*F,F,F-F-/F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,& %\"xG\"\"\"F+!\"\"F+F*F+,&F*F+F+F+F+-F$6#F)F,/F*F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G-%&LimitG6$*&%\"xG\"\"\",&F)F*F*F*F*/F)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Limit((x^3-x)/(x-1),x=-1):\neval_lim(%,info=true):" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&*$)%\"xG\"\"$\"\"\"F,F *!\"\"F,,&F*F,F,F-F-/F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"! " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 312 "p1 := plot(x^2+x,x=-3..2.5):\np2 := plot([[[1,2]]$2] ,style=point,symbol=[circle$2],symbolsize=[15,18],color=brown):\nt1 := plots[textplot]([[2.5,-.2,`x`],[-.15,6.5,`y`]],color=COLOR(RGB,.01,0, 0)):\nplots[display]([p||(1..2),t1],view=[-3..2.5,-.5..6.5],labels=[`` ,``],tickmarks=[4,3],\n title=\"graph of y = f(x)\");" }}{PARA 13 " " 1 "" {GLPLOT2D 358 330 330 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!\"$\"\"! $\"\"'F*7$$!3_LL3_c6!)G!#<$\"3br?H<0&\\T&F07$$!3um\"z>#\\!ex#F0$\"3K,Y !H/)GH\\F07$$!3OL$ekf'\\eEF0$\"3Y3A#pb2\"4WF07$$!3IL$3_n5/a#F0$\"3XET] BdF8RF07$$!3jm\"Hd*f)GU#F0$\"3%)[8d_0\\ZMF07$$!3HLektb#RJ#F0$\"3nljs') fKSIF07$$!3F+D\"yO.6?#F0$\"3xII)zm_Pk#F07$$!3SLe9\"[AW3#F0$\"3Y=.w)f%R gAF07$$!31+Dc\"z:\"o>F0$\"3/'=Nu(RO0>F07$$!3kmmTl+[[=F0$\"3OPh#o[)Ro:F 07$$!3FL$3F&[5Vv)= Fio7$$!3!omm;&4Qd5F0$\"3oCR3GDNng!#>7$$!35,]7`-Q_$*Fio$!3#ya2A8'ycgF^q 7$$!3Armm\"R37F)Fio$!3Tb*Gec>*H9Fio7$$!3#***\\7.:*\\3(Fio$!37b'*=V5Gl? Fio7$$!3C)*\\P4DVbfFio$!3VRH+s[r3CFio7$$!3slm\"z>cox%Fio$!3Mz\"fVo?]\\ #Fio7$$!3El;HK_b%p$Fio$!3E$HYeR\"eHBFio7$$!31OL3Fu9FDFio$!3cV>E:+]))=F io7$$!3tO$e9\")QXJ\"Fio$!322`x#eP<9\"Fio7$$!3c++voWf*e#F^q$!3h%>COZMD_ #F^q7$$\"3$4nmT&=&4\"))F^q$\"3;,6$*z0G(e*F^q7$$\"3u%****\\<$))e?Fio$\" 3o+)Hy;$y#[#Fio7$$\"3S&**\\P%*=6@$Fio$\"33/5%3VZAC%Fio7$$\"3z'*\\(oHmf K%Fio$\"3qL[/PZO(>'Fio7$$\"3M(**\\i?9Qc&Fio$\"35EpTeqTf')Fio7$$\"3[HLL 3c2wmFio$\"3El^6;uI86F07$$\"3E(***\\i?ljyFio$\"3M/1&QWNZS\"F07$$\"3:k; aQGxR*)Fio$\"3;R9!zmsJp\"F07$$\"3Y++v3di65F0$\"3CTRN$G7].#F07$$\"3\"G$ 3-3PKA6F0$\"3Gkkg8U$>Q#F07$$\"3++vV8].Q7F0$\"3!3pMzql2x#F07$$\"3)GL$ek R;^8F0$\"3)H)=xB!3o<$F07$$\"3I*\\(=#\\*fp9F0$\"3&\\DD'f@KHOF07$$\"33nm m\")pm$e\"F0$\"3#)yy[!4o;4%F07$$\"3dm;zkuJ+#fO1PlQ6&F07$$\"3T++]F0$\"32culPsb " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x) = x/(x^2- x);" "6#/-%\"fG6#%\"xG*&F'\"\"\",&*$F'\"\"#F)F'!\"\"F-" }{TEXT -1 12 " , find (a) " }{XPPEDIT 18 0 "Limit(f(x),x = 1);" "6#-%&LimitG6$-%\"fG 6#%\"xG/F)\"\"\"" }{TEXT -1 11 " and (b) " }{XPPEDIT 18 0 "Limit(f(x ),x = 0);" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Limit(x/(x^2-x),x=1):\neval_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&%\"xG\"\"\",&*$)F'\"\"#F(F(F'!\"\"F-/ F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%&LimitG6%*&%\"xG\"\"\",&*$) F(\"\"#F)F)F(!\"\"F./F(F)%%leftG,$%)infinityGF.%+~~while~~~G/-F%6%F'F/ %&rightGF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Iso~the~(two-sided)~lim it~does~not~exist.G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Limit(x /(x^2-x),x=0):\neval_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&%\"xG\"\"\",&*$)F'\"\"#F(F(F'!\"\"F-/F'\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(-F$6#%\"xG\"\"\"F+!\" \",&F+F,F,F-F-/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&Limit G6$*&\"\"\"F),&%\"xGF)F)!\"\"F,/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 388 "p1 := plot(1/(x-1),x=-2..3.5,y=-3..2.9,discon t=true):\np2 := plot([[[0,-1]]$2],style=point,symbol=[circle$2],symbol size=[15,18],color=brown):\np3 := plot([[[1,-3],[1,3.5]]],color=black, linestyle=3):\nt1 := plots[textplot]([[3.5,-.15,`x`],[-.15,2.9,`y`]],c olor=COLOR(RGB,.01,0,0)):\nplots[display]([p||(1..3),t1],view=[-2..3.5 ,-3..2.9],labels=[``,``],\n title=\"graph of y = f(x)\"); " }}{PARA 13 "" 1 "" {GLPLOT2D 358 330 330 {PLOTDATA 2 "6+-%'CURVESG6% 7gn7$$!\"#\"\"!$!3:LLLLLLLL!#=7$$!3l)=(eP&3Y$>!#<$!30hYNB&4wS$F-7$$!3/ ^!)yvZaV0@&o\"F1$!3nq@MhF5CPF-7$$!3xXF-7$$!3x\"\\q')4J@7\"F1$!3c()H@ =UC7ZF-7$$!3_-8(=Bt_1\"F1$!3W>t*HMu>%[F-7$$!3s#[k$)GPm(**F-$!3yHmP()F-$!3]c=#z7XoL&F- 7$$!3udF!e'=$z9)F-$!3Y]e\"3vp-^&F-7$$!3[fa\"HZ/4](F-$!3P%Qa9R!*Rr&F-7$ $!3ydpB'Q\"y%)oF-$!3P,HnDI\\AfF-7$$!34.T%3e@>C'F-$!3]w&o%ep!p:'F-7$$!3 =J7Q+cd^cF-$!3-%='*o6L\"*Q'F-7$$!3td-[2x![,&F-$!3#GYqP&>4gmF-7$$!3Ya@6 F[Q`VF-$!31`iDkx*p'pF-7$$!3CZ)z`E9wx$F-$!3?Hema1:esF-7$$!3[y*e\\Gmd:$F -$!3%>\\4FfH7g(F-7$$!3x_bJtkL8DF-$!39_B)Qvt9*zF-7$$!3GUNlOU%[)=F-$!3!G .\"Rfu29%)F-7$$!3iRV@\"oXnF\"F-$!3M!f^'>m!y'))F-7$$!3JJfkL\"fb,'!#>$!3 Lli-5xdK%*F-7$$\"3mX+\\EL7K^!#@$!3A!zaeZ80+\"F17$$\"3)*oziu4,HlFbu$!31 JF\\n1&)p5F17$$\"3Fj,Vlg()R7F-$!3s/!37OO:9\"F17$$\"35WX#eu%f\")=F-$!3T Ocm4!pMCz =+&F-$!3KT`r6#G2+#F17$$\"3iteJG_4QcF-$!3+JS29kd#H#F17$ $\"3KIXs3>5piF-$!3#3JxIz>.o#F17$$\"3)RU,U^J*[oF-$!3Um<]3n_tJF17$$\"3') e`\"en\"[8vF-$!3FvLdNwo@SF17$$\"3s\\G_iK'y5)F-$!3ue(\\cII]G&F17$$\"3eo yU0!)fT()F-$!3o$>*HgieYzF17$$\"3-#\\t7^!)[/*F-$!351&*[?%*)p/\"!#;7$$\" 3Y:\">r,j\"[$*F-$!36$fWI!f7M:Ffz7$$\"3)oLRGDA6^*F-$!3'zq=A;,b/#Ffz7$$ \"3Ie&f&)[\"3u'*F-$!3W?e!yh^#oIFfz7$$\"3+p'>k5hbv*F-$!3K_9r*)>+\"4%Ffz 7$$\"3rz(zUsSq$)*F-$!3&[JyC[-l8'Ffz7$$\"3_M)4K`!yx)*F-$!3qT$>0k-?=)Ffz 7$$\"3J*))R@M?&=**F-$!32FCC&>+tA\"!#:7$$\"3r;\\gY-*)Q**F-$!3G\"pUD***R O;Fj\\l7$$\"37W*p5:g#f**F-$!3G(zsa=*faCFj\\l7$$\"3_r\\`b+jz**F-$!3'oY/ 4b$>4\\Fj\\l7$$\"2++++'********F1$!3m*)eJ,+++D!\"*7gn7$$\"3\"******p++ ++\"F1$\"3TvlJY9dG9F_^l7$$\"3r-Xm4Hq,5F1$\"3]@^g+@IseFj\\l7$$\"3G0!HB \"eS.5F1$\"3'R<.]3dh$HFj\\l7$$\"323N*\\r3^+\"F1$\"3'H&pr(RRu&>Fj\\l7$$ \"3'3,ewh6o+\"F1$\"3e&H\"=^+3o9Fj\\l7$$\"3A;q)HU<-,\"F1$\"3w<]$Qp.sy*F fz7$$\"3f@gJGKi85F1$\"3Z-=hFSSStFfz7$$\"3JKS(*Q[V?5F1$\"39[=BBNg$*[Ffz 7$$\"3EV?j\\kCF5F1$\"3yHHtcH?qOFfz7$$\"3$\\1[4np3/\"F1$\"3a(p%p!=-oW#F fz7$$\"3g'3kA*G\\a5F1$\"3X7$)49<5N=Ffz7$$\"3U!H7j()*>y5F1$\"3#Q(R8,Wxy 7Ffz7$$\"3,%\\g.'o!>5\"F1$\"37Qo6U?)G\")*F17$$\"3%e-dj#)G_:\"F1$\"3]Vd MLC5UkF17$$\"3lMn&\\C/*37F1$\"33@<0W@)oy%F17$$\"3cC8m`XKi7F1$\"3vy$[Cn r?\"QF17$$\"3'4)f5=?&=J\"F1$\"3XM&G=7\\m?$F17$$\"3GAdzQ[8j8F1$\"3Or3\" )>!)z`FF17$$\"3>')QJ0;<;9F1$\"3BOw[X\\&GS#F17$$\"3w#>\\AGQ!p9F1$\"3%zb v^8A?8#F17$$\"3Mixhd\"=M_\"F1$\"3d([N)f$=0\">F17$$\"3#>)>W\")fJr:F1$\" 3_jayu[M]**G>Z\"F17$$\"37R;qBubJ\"F17$$\"3144'Gm3I)=F1 $\"3jDTq?<\\K6F17$$\"3C@You\"G&Q>F1$\"3G\"fLR6)\\l5F17$$\"3g]y1]Bn()>F 1$\"3**QJr<:[75F17$$\"3LWgR*H\"fT?F1$\"3M\"pC]j%p+'*F-7$$\"3g#Gx)*)[$H 4#F1$\"3mMmqzen\\\"*F-7$$\"3;:\"[-a1l9#F1$\"3%[;/\"Q#[@s)F-7$$\"3#y'ya Q?q&>#F1$\"3oZ:+tvGj$)F-7$$\"3s@^aHgw[AF1$\"3gwBv9_!z+)F-7$$\"3b*GGGg% )QI#F1$\"3=0#GtN!RpwF-7$$\"3[wA!H[l=N#F1$\"35RMN'o&=(R(F-7$$\"3#eMhy9' o.CF1$\"3e2Py`'*4CrF-7$$\"3Wy(>a'>AdCF1$\"3]Oa()RDPioF-7$$\"3CI\")R=jf 4DF1$\"3NV%>a[(GCmF-7$$\"3qSP%z>r-c#F1$\"3yU5)R@U\"4kF-7$$\"3O'p[=,Plh #F1$\"3sM!)\\DJ1'='F-7$$\"3)*oa;[V4nEF1$\"3#zr\"fT2Y)*fF-7$$\"3t*)4$f4 v5s#F1$\"3UBln*)*pFF1$\"3%*\\+9m/v\\cF-7$$\"3z%HWJB mM#GF1$\"3_KTQq41%[&F-7$$\"3KR#*R0NytGF1$\"3iW25JezO`F-7$$\"3SP'oJbzj# HF1$\"3'[/n&)*\\3\">&F-7$$\"3I)[XN!=!y(HF1$\"3/#fiUN=h0&F-7$$\"3')=R7M hjJIF1$\"3R72yR79A\\F-7$$\"3m3H$G*\\[$3$F1$\"3caz%>g]'*z%F-7$$\"3SW%f& Rz]OJF1$\"3)>J\"f6\\`!o%F-7$$\"3#oP;&H=4*=$F1$\"3e^le\"R/\"oXF-7$$\"3Y \\_Bj4TPKF1$\"3/\"zedf^%pWF-7$$\"3q?&)GV,z#H$F1$\"3!G#)HQ;)\\hVF-7$$\" 3\"\\;3@%>KULF1$\"3q*R(p^zEpUF-7$$\"3'p711L^<%F-7$$\" 31&fR[D!oXMF1$\"3cv@*3%>%))3%F-7$$\"3++++++++NF1$\"3A+++++++SF--%'COLO URG6&%$RGBG$\"*++++\"!\")$F*F*F_am-F$6&7#7$F_am$!\"\"F*-%'SYMBOLG6$%'C IRCLEG\"#:-Fi`m6&F[am$\")#)eqkF^am$\"))eqk\"F^amF_bm-%&STYLEG6#%&POINT G-F$6&Fbam-Fgam6$Fiam\"#=F[bmFabm-F$6%7$7$$\"\"\"F*$!\"$F*7$F^cmFd`m-F i`m6&F[amF*F*F*-%*LINESTYLEG6#\"\"$-%%TEXTG6%7$$\"#NFeam$Fj\\lF)Q\"x6 \"-%&COLORG6&F[am$F_cmF)F_amF_am-Fjcm6%7$F_dm$\"#HFeamQ\"yFadmFbdm-%&T ITLEG6#Q3graph~of~~y~=~f(x)Fadm-%+AXESLABELSG6%%!GFcem-%%FONTG6#%(DEFA ULTG-%%VIEWG6$;F(F]dm;F`cmFidm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp le 6 " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Limit((2*x ^2-x-3)/(x+1),x=-1)" "6#-%&LimitG6$*&,(*&\"\"#\"\"\"*$%\"xGF)F*F*F,!\" \"\"\"$F-F*,&F,F*F*F*F-/F,,$F*F-" }{TEXT -1 15 " if it exists. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Limit((2*x^2-x-3)/(x+1),x=-1):\neval_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,(*&\"\"#\"\"\")%\"xGF)F*F*F,! \"\"\"\"$F-F*,&F,F*F*F*F-/F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G -%&LimitG6$*(,&%\"xG\"\"\"F+F+F+,&*&\"\"#F+F*F+F+\"\"$!\"\"F+-F$6#F)F0 /F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&*&\"\"#\"\" \"%\"xGF+F+\"\"$!\"\"/F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G!\"& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "p1 := plot((2*x^2-x-3)/(x+1 ),x=-2.5..2.5):\np2 := plot([[[-1,-5]]$2],style=point,symbol=[circle$2 ],symbolsize=[15,18],color=brown):\nt1 := plots[textplot]([[2.5,-.23,` x`],[-.13,2.8,`y`]],color=COLOR(RGB,.01,0,0)):\nplots[display]([p1,p2, t1],view=[-2.5..2.5,-7..2.8],labels=[``,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 365 381 381 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$!3++++++++D!#<$! \")\"\"!7$$!3smm;HU,\"R#F*$!3cKLLe%G?y(F*7$$!3=L$3FH'='H#F*$!3[lmT&esB f(F*7$$!3gmmTgBa*=#F*$!3MKL$3s%3ztF*7$$!3wmm\"H_\">#3#F*$!3_LL$e/$QkrF *7$$!3ML$3_!4Nv>F*$!3cnmT5=q]pF*7$$!3km;/wfHw=F*$!3uLL3_>f_nF*7$$!3;+] PM.ttc \"F*$!3!))**\\P*o%Q7'F*7$$!3;LL$epjJX\"F*$!3Kmmm\"RFj!fF*7$$!3amm\"z/o tN\"F*$!3vJL$e4OZr&F*7$$!3))****\\P[_\\7F*$!33,++v'\\!*\\&F*7$$!3#**** *\\7)Q79\"F*$!3u'****\\ixCG&F*7$$!3'*****\\i^)o.\"F*$!3e-++D.xt]F*7$$! 3vlmT50A@%*!#=$!3#*QL3-TC%)[F*7$$!3OKLLeaR%H)Fap$!3/mmm\"4z)eYF*7$$!3k JLLLo#)RtFap$!3Mmmmm`'zY%F*7$$!3f***\\PfO%HiFap$!3O++v=t)eC%F*7$$!3/ML L$3`lC&Fap$!3Enmm;1J\\SF*7$$!3q'**\\P4u\"oTFap$!3y***\\(=[jLQF*7$$!3*z **\\7G-89$Fap$!3g***\\iXg#GOF*7$$!3%)GL$3Fp)p?Fap$!3xlm;aQ(RT$F*7$$!3Y KL3-$ff3\"Fap$!3\\mmTg=>px5Fap$!3#GL$3-;Y%y#F*7$$\"3w++v$f4t.#Fap$!3%)***\\73QDf#F*7$ $\"3OPL$e*GstIFap$!3_KL$3Ub_Q#F*7$$\"3Y+++]#RW9%Fap$!3!*******\\@6r@F* 7$$\"3Y,+]7j#>>&Fap$!3q****\\PZhh>F*7$$\"3t-+D1RU0iFap$!3Y***\\(=_\"*e &Q`\"F*7$$\"3:jmm;p)=M)Fap$!39nmm;Ei J8F*7$$\"3O-++v=]@%*Fap$!3J*****\\i*p:6F*7$$\"35L$e*[$z*R5F*$!3,RL$3-8 /?*Fap7$$\"3e++]iC$p9\"F*$!3Y*)****\\2NhqFap7$$\"3[m;H2qcZ7F*$!3Spm;a) f'[]Fap7$$\"3O+]7.\"fFN\"F*$!3x#***\\Pz\"[%HFap7$$\"3Ymm;/Ogb9F*$!3![r mm;z#z))!#>7$$\"3w**\\ilAFj:F*$\"3['***\\7`Wl7Fap7$$\"3yLLL$)*ppm\"F*$ \"3wummm'*RRLFap7$$\"3)RL$3xe,t#[(>F*$\"3a4+++&Qk\\*Fap7$$\"3SnmT&G!e&3#F *$\"3-NL$3dg6<\"F*7$$\"3#RLLL)Qk%=#F*$\"3inmmmxGp8F*7$$\"37+]iSjE!H#F* $\"3C++D\"oK0e\"F*7$$\"3a+]P40O\"R#F*$\"3M,+v=5s#y\"F*7$$\"3++++++++DF *$\"\"#F--%'COLOURG6&%$RGBG$\"#5!\"\"$F-F-Fb[l-F$6&7#7$$Fa[lF-$!\"&F-- %'SYMBOLG6$%'CIRCLEG\"#:-F\\[l6&F^[l$\")#)eqkF,$\"))eqk\"F,Fc\\l-%&STY LEG6#%&POINTG-F$6&Fe[l-F[\\l6$F]\\l\"#=F_\\lFe\\l-%%TEXTG6%7$$\"#DFa[l $!#B!\"#Q\"x6\"-%&COLORG6&F^[l$\"\"\"Ff]lFb[lFb[l-F_]l6%7$$!#8Ff]l$\"# GFa[lQ\"yFh]lFi]l-%+AXESLABELSG6%%!GFi^l-%%FONTG6#%(DEFAULTG-%%VIEWG6$ ;$!#DFa[lFb]l;$!\"(F-Fc^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }}{PARA 0 " " 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Limit((x^3-8)/(x-2),x = 2); " "6#-%&LimitG6$*&,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F+,&F)F+\"\"#F-F-/F)F/ " }{TEXT -1 16 " if it exists. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Limit((x^3-8)/(x-2),x=2):\ne val_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*& ,&*$)%\"xG\"\"$\"\"\"F,\"\")!\"\"F,,&F*F,\"\"#F.F./F*F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,&%\"xG\"\"\"\"\"#!\"\"F+,(*$)F *F,F+F+*&F,F+F*F+F+\"\"%F+F+-F$6#F)F-/F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,(*$)%\"xG\"\"#\"\"\"F-*&F,F-F+F-F-\"\"% F-/F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"#7" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "p1 := plot((x^3-8)/(x-2),x=-2.5..3.5):\np2 := p lot([[[2,12]]$2],style=point,symbol=[circle$2],symbolsize=[15,18],colo r=brown):\nt1 := plots[textplot]([[3.5,-.7,`x`],[-.17,23,`y`]],color=C OLOR(RGB,.01,0,0)):\nplots[display]([p||(1..2),t1],view=[-2.5..3.5,-.7 ..23],labels=[``,``],\n title=\"graph of y = f(x)\");" }}{PARA 13 " " 1 "" {GLPLOT2D 305 378 378 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!3+++++++ +D!#<$\"3+++++++]_F*7$$!3!******\\2<#pBF*$\"3qb:Z)RbZ([F*7$$!3#)***\\7 bBaD#F*$\"3ygJ.$H)3wXF*7$$!36++]K3XF@F*$\"3oV\\qz`9rUF*7$$!3%)****\\F) H')*>F*$\"3mzEPK:E(*RF*7$$!3#****\\i3@/(=F*$\"3YH')QnGjdPF*7$$!3;++Dr^ b^iUuFao$\"3/=i!\\#=SlIF*7$$!3B++]7YY0jFao$\"3&y*[H#z2QFao$\"3o3QQuVV$ Q$F*7$$!3S++]7RKvCFao$\"3eVz@-v?mNF*7$$!3s,+++P'eH\"Fao$\"3!ox*G())>wv $F*7$$!3'\\q)***\\7*3=!#@$\"3KW;A]#Q'**RF*7$$\"3_,+]isVI7Fao$\"3K&[p5@ F7E%F*7$$\"3&=++](o:;DFao$\"3U)4'>H=amXF*7$$\"3#>++v$)[op$Fao$\"39^Kx! oOg([F*7$$\"3W*****\\i%Qq\\Fao$\"3^jV?eT7T_F*7$$\"3]****\\(QIKH'Fao$\" 3]L:gkNpacF*7$$\"38****\\7:xWuFao$\"3p:LBJ0?VgF*7$$\"3E,++vuY)o)Fao$\" 3cH8kl\")e#\\'F*7$$\"3!z******4FL(**Fao$\"3_VfVazL*)pF*7$$\"3#)****\\d 6.B6F*$\"3P*y:dHhs](F*7$$\"3(****\\(o3lW7F*$\"3eH,eAvXQ!)F*7$$\"3!**** *\\A))oz8F*$\"3gO9$>*)=Hm)F*7$$\"3e******Hk-,:F*$\"3?V&eN?L^D*F*7$$\"3 6+++D-eI;F*$\"3x^5;?B&*>**F*7$$\"3u***\\(=_(zu\"F*$\"3/6kG!yO^0\"!#;7$ $\"3M+++b*=j(=F*$\"3JH#*3t5KF6Fcv7$$\"3g***\\(3/3(*>F*$\"3_28l(4\\#)> \"Fcv7$$\"33++vB4JB@F*$\"3S3;k7r]v7Fcv7$$\"3u*****\\KCnC#F*$\"3_qYD%)= 7a8Fcv7$$\"3s***\\(=n#fP#F*$\"399q.@\")oR9Fcv7$$\"3P+++!)RO+DF*$\"3FW \"[#*za__\"Fcv7$$\"30++]_!>wi#F*$\"3'**41!**>'fh\"Fcv7$$\"3O++v)Q?Qv#F *$\"3d?&*4^n64Fcv7$$\"3++++gEd@JF*$\"3wZf;>ht)*>Fcv7$$\"39++v3'>$[KF* $\"3Rv*3)\\>#[5#Fcv7$$\"37++D6EjpLF*$\"3t\\*Hd\"*o$4AFcv7$$\"3++++++++ NF*$\"3+++++++DBFcv-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-F$6&7#7$ $\"\"#Fb[l$\"#7Fb[l-%'SYMBOLG6$%'CIRCLEG\"#:-F[[l6&F][l$\")#)eqk!\")$ \"))eqk\"Fd\\lFe\\l-%&STYLEG6#%&POINTG-F$6&Fe[l-F\\\\l6$F^\\l\"#=F`\\l Fg\\l-%%TEXTG6%7$$\"#NF`[l$!\"(F`[lQ\"x6\"-%&COLORG6&F][l$\"\"\"!\"#Fa [lFa[l-Fa]l6%7$$F*F_^l$\"#BFb[lQ\"yFi]lFj]l-%&TITLEG6#Q3graph~of~~y~=~ f(x)Fi]l-%+AXESLABELSG6%%!GF^_l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#DF`[l Fd]l;Ff]lFd^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Limit((x^2-5*x+4)/(x^2-2*x-8),x=4) " "6#-%&LimitG6$*&,(*$%\"xG\"\"#\"\"\"*&\"\"&F+F)F+!\"\"\"\"%F+F+,(*$F )F*F+*&F*F+F)F+F.\"\")F.F./F)F/" }{TEXT -1 16 " if it exists. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Limit((x^2-5*x+4)/(x^2-2*x-8),x=4):\neval_lim(%,info=true):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,(*$)%\"xG\"\"#\"\"\"F,*& \"\"&F,F*F,!\"\"\"\"%F,F,,(F(F,*&F+F,F*F,F/\"\")F/F//F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**-F$6#,&%\"xG\"\"\"\"\"%!\"\"F- ,&F,F-F-F/F-F+F/,&F,F-\"\"#F-F//F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G-%&LimitG6$*&,&%\"xG\"\"\"F+!\"\"F+,&F*F+\"\"#F+F,/F*\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 353 "p1 := plot( (x-1)/(x+2),x=-8..6.9,y=-4..4.9,discont=true):\np2 := plot([[[4,1/2]]$ 2],style=point,symbol=[circle$2],symbolsize=[15,18],color=brown):\np3 \+ := plot([[[-2,-4],[-2,4.9]]],color=black,linestyle=3):\nt1 := plots[te xtplot]([[6.9,-.15,`x`],[-.4,4.9,`y`]],color=COLOR(RGB,.01,0,0)):\nplo ts[display]([p||(1..3),t1],view=[-8..6.9,-4..4.9],labels=[``,``]);" }} {PARA 13 "" 1 "" {GLPLOT2D 370 267 267 {PLOTDATA 2 "6*-%'CURVESG6%7gn7 $$!\")\"\"!$\"3++++++++:!#<7$$!3qa([`2<#pyF-$\"3p7!)[G996:F-7$$!3Y.A!> bBav(F-$\"3jy^/wuC@:F-7$$!34kM\\L3XFwF-$\"325A@f55L:F-7$$!3!o)p$)G)H') \\(F-$\"39.$eSY!fX:F-7$$!3'p()Gz3@/P(F-$\"3i4S69ahe:F-7$$!3)H&eCt^b^sF -$\"3uj#)*4Nf7d\"F-7$$!34jSd.kZGrF-$\"37\">**e0p\\e\"F-7$$!3&y\\8R;)=, qF-$\"3KMR?au&)*f\"F-7$$!3![%=DR\"3V(oF-$\"3B'4gK'>Z:;F-7$$!33w)\\$Qkz VnF-$\"3^$*[Mw[SK;F-7$$!3g@k:h;%)GmF-$\"3)e+Y\"4.6[;F-7$$!37?:+4)H%*\\ 'F-$\"3[/?6]6vm;F-7$$!3_N![$zl[pjF-$\"3cc,R?&zlo\"F-7$$!3%p'>o*>iUC'F- $\"3F#3UBjOoq\"F-7$$!3Q4_BmkaIhF-$\"3&*48,^hHEYSl#=F-7$$!3O%=;z*3=+bF-$\"3au>!zb) 4d=F-7$$!3#GyW2GcpP&F-$\"3]y3$H&RP))=F-7$$!3UTw$)>VQ[_F-$\"3oF -7$$!36$\\-R7:.8&F-$\"3w#*HOm*p$e>F-7$$!3:-@\\X:'H+&F-$\"3ayht\"z8!**> F-7$$!3%3'[fppnq[F-$\"35%\\>Jm\\]/#F-7$$!3#)Q>SdG_bZF-$\"3Gf1\\'fA()3# F-7$$!3j!f$[hK:JYF-$\"3J[*fpV%=S@F-7$$!3X?iK*HnE]%F-$\"3cG)[31@()>#F-7 $$!3u;9;_)opP%F-$\"3aoOM;;6iAF-7$$!3%\\tN79\\`D%F-$\"3v?I+!*4%pQF-$\"3J:8^'*fx/EF-7$$!3aLzd#zC?v$F-$\"32\\kFOXI7FF-7$$!3q#=qm 0\"oBOF-$\"3]j%ey]`w%GF-7$$!3u9AC.'>H]$F-$\"3)zi?zv9h*HF-7$$!3aG)y&)3* owLF-$\"3s_\"\\NET\"zJF-7$$!3'4$zl(ovKD$F-$\"3oK#)4=rs$R$F-7$$!3QrCD%H tS7$F-$\"3U&zdM6l)oOF-7$$!31.VLLgj**HF-$\"3Z!*pb(H#4,SF-7$$!3g]O(=lz/-&F-7$$!3#*G%>Vq8-j#F -$\"3'H3s'3+HgdF-7$$!3ubQH\"!#;7$$!3U-1\\0R-\">#F- $\"3'[.dq1%[q;F^z7$$!31`BS/uOI@F-$\"38CEU;()=,CF^z7$$!3ok]OiF^z7$$!3P)e+J'=fK?F-$\"3oH7fw9v/$*F^z7$$!3;TaK,RWC?F-$\"3jK M/'***HP7!#:7$$!3'RH]&RfH;?F-$\"3=^#f:R\\4&=Fb\\l7$$!3e?Fme>A7?F-$\"3S _\\S&Q)fkCFb\\l7$$!3?Z^xxz93?F-$\"3kYw3o^*=p$Fb\\l7$$!3Ptv)o*R2/?F-$\" 3+n_4wexttFb\\l7$$!3*******f,+++#F-$\"3KF-$!3PV%)[Dr&G9#F\\^l7$$!3)=$p]_w$R*>F-$!3_C?g$4y&Q\\Fb\\l7$$!3 1kQ,>`(y)>F-$!3G+ty.>HkCFb\\l7$$!3A'z?b)H\"=)>F-$!3N>]uq&G&R;Fb\\l7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "Limit((sqrt(x+2)-sqrt(2))/x,x=0):\neval_lim(%, info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&*$,&%\"x G\"\"\"\"\"#F+#F+F,F+*$F,F-!\"\"F+F*F//F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,&*$,&%\"xG\"\"\"\"\"#F-#F-F.F-*$F.F/! \"\"F--F$6#,&F*F-F0F-F-F,F1F4F1/F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(-F$6#%\"xG\"\"\"F+!\"\",&*$,&F+F,\"\"#F,#F,F1F, *$F1F2F,F-/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$* &\"\"\"F),&*$,&%\"xGF)\"\"#F)#F)F.F)*$F.F/F)!\"\"/F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"%!\"\"\"\"##\"\"\"F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "p1 \+ := plot((sqrt(x+2)-sqrt(2))/x,x=-2..3.5,discont=true):\np2 := plot([[[ 0,sqrt(2)/4]]$2],style=point,\n symbol=[circle$2],symbolsize= [15,18],color=brown):\np3 := plot([[[-2,sqrt(2)/2]]$4],style=point,sym bol=[circle$2,diamond,cross],color=red,\n symbolsize=[15,1 0$3]):\nt1 := plots[textplot]([[3.5,-.03,`x`],[-.15,.79,`y`]],color=CO LOR(RGB,.01,0,0)):\nplots[display]([p||(1..3),t1],view=[-2..3.5,-.03.. 0.79],labels=[``,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 523 193 193 {PLOTDATA 2 "6--%'CURVESG6$7en7$$!\"#\"\"!$\"3sva'=\"y1rq!#=7$$!3K3_DG s]#*>!#<$\"39$yG4%oAjmF-7$$!3k;/^cW,&)>F1$\"3;#Qwrj^x]'F-7$$!3'\\ilZo@ v(>F1$\"3qjqHVWG$R'F-7$$!3GL3-8*G+(>F1$\"3I&>#\\fn')*H'F-7$$!3!*\\7`pL /b>F1$\"3_?;s779\\hF-7$$!3am;/Ey0S>F1$\"3L-xIrfcFgF-7$$!3#)*\\i!Rn35>F 1$\"3%=sJ])42MeF-7$$!3ILL3_c6!)=F1$\"3GxV-ASM!o&F-7$$!3\"*\\7.(Ggz#=F1 $\"3SUr^jw\\naF-7$$!3um\"z>#\\!ex\"F1$\"3gXH!*=^V(H&F-7$$!3OL$ekf'\\e; F1$\"3:rL^W7^.]F-7$$!3IL$3_n5/a\"F1$\"3#GXX#3jyzZF-7$$!3jm\"Hd*f)GU\"F 1$\"3([]n>!3/+YF-7$$!3HLektb#RJ\"F1$\"3WrZn'f*GfWF-7$$!3/+D\"yO.6?\"F1 $\"3\")>d*z16FL%F-7$$!3SLe9\"[AW3\"F1$\"3i8KZJ@]/-%F-7$$!3kKL3F&[5V(F-$\"3J 44]AXEWRF-7$$!3%))***\\7KxWiF-$\"3;'4Q%f4UlQF-7$$!3--+]Ppi`]F-$\"3?OC( y&H_#z$F-7$$!3U++](yOd!RF-$\"3(H]7e3'QFPF-7$$!3AN$e9cUL'GF-$\"3qGTzrp. sOF-7$$!3:nm;/]$Qi\"F-$\"3_yHch!o.h$F-7$$!32zmmm^4Qd!#>$\"3[\"pN>Qj7c$ F-7$$\"3/*)*\\(ou>wkFds$\"3`O>8/rO2NF-7$$\"3zGLL3;zG`T$F-7$$\"3w,]i!\\nX/%F-$\"31Ak#)3!eGP$ F-7$$\"3HML3-Q9B_F-$\"3uKW10knILF-7$$\"3tM$3xwWaI'F-$\"31AAe,&*o$H$F-7 $$\"3%Rm;Hd_GZ(F-$\"3C?\\X$f=bD$F-7$$\"3Ej;a)=hao)F-$\"3(G0$)))=:w@$F- 7$$\"3%***\\7`0/T(*F-$\"3S6gS(oef=$F-7$$\"3qm;a=&4\")3\"F1$\"3%4*GN#*o 1`JF-7$$\"3Z****\\<$))e?\"F1$\"3WKR_2US?JF-7$$\"3a**\\P%*=6@8F1$\"3!\\ L%Qmol*3$F-7$$\"3o*\\(oHmfK9F1$\"3Iqiw>!o41$F-7$$\"3u**\\i?9Qc:F1$\"3t (p5\\ac-.$F-7$$\"3%HLL3c2wm\"F1$\"30X/unAi.IF-7$$\"3s***\\i?ljy\"F1$\" 3?!=)fom7wHF-7$$\"3SmT&QGxR*=F1$\"3;!3\"*G[.?&HF-7$$\"3Y++v3di6?F1$\"3 @;$RWzUk#HF-7$$\"3\"G$3-3PKA@F1$\"3D*Q@@([7.HF-7$$\"3++vV8].QAF1$\"3-v ?]8\"p%zGF-7$$\"3)GL$ekR;^BF1$\"37f2#)GR,dGF-7$$\"3I*\\(=#\\*fpCF1$\"3 #GB]Z*3=MGF-7$$\"33nmm\")pm$e#F1$\"3]aGH#45G\"GF-7$$\"3dm;zkuJ+FF1$\"3 K\"QshA`:z#F-7$$\"3OL3-B?+;GF1$\"3%Rzg@`S5x#F-7$$\"3T++] " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 10 " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Limit((sqrt(x+1)-2)/(x-3),x=3)" "6# -%&LimitG6$*&,&-%%sqrtG6#,&%\"xG\"\"\"F-F-F-\"\"#!\"\"F-,&F,F-\"\"$F/F //F,F1" }{TEXT -1 15 " if it exists. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Limit((sqrt(x+1)-2)/(x-3 ),x=3):\neval_lim(%,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%& LimitG6$*&,&*$,&%\"xG\"\"\"F+F+#F+\"\"#F+F-!\"\"F+,&\"\"$F.F*F+F./F*F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,&*$,&%\"xG\"\"\" F-F-#F-\"\"#F-F/!\"\"F--F$6#,&F/F-F*F-F-,&\"\"$F0F,F-F0F3F0/F,F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(-F$6#,&\"\"$!\"\"%\"x G\"\"\"F/F+F-,&\"\"#F/*$,&F.F/F/F/#F/F1F/F-/F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&\"\"\"F),&\"\"#F)*$,&%\"xGF)F)F)#F)F+F )!\"\"/F.\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"%" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 425 "p1 := plot((sqrt(x+1)-2)/(x-3),x=-1..4.5,discont=true):\np2 := \+ plot([[[3,1/4]]$2],style=point,\n symbol=[circle$2],symbolsiz e=[15,18],color=brown):\np3 := plot([[[-1,1/2]]$4],style=point,symbol= [circle$2,diamond,cross],color=red,\n symbolsize=[15,10$3] ):\nt1 := plots[textplot]([[4.5,-.03,`x`],[-.13,.59,`y`]],color=COLOR( RGB,.01,0,0)):\nplots[display]([p||(1..3),t1],view=[-1..4.5,-.03..0.59 ],labels=[``,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 523 193 193 {PLOTDATA 2 "6--%'CURVESG6$7en7$$!\"\"\"\"!$\"3++++++++]!#=7$$!3=$3_DG s]#**F-$\"3Gd/^Bad#z%F-7$$!3OmT5lX9])*F-$\"3!)f\\!Q)Gh6ZF-7$$!3a\\ilZo @v(*F-$\"3G%4wZ?>8l%F-7$$!3sK$3-8*G+(*F-$\"3Kw_#HJv;g%F-7$$!33*\\7`pL/ b*F-$\"3V9)4RyM2_%F-7$$!3bmmTg#y0S*F-$\"3#>l3i]yYX%F-7$$!3R+]i!Rn35*F- $\"3WG\"*)*\\%*4[VF-7$$!36LL$3_c6!))F-$\"3!f&4OsF8iUF-7$$!3?+DJqGgz#)F -$\"3[F*zKlp69%F-7$$!3Gn;z>#\\!exF-$\"3Q04mLm'G/%F-7$$!3iLLekf'\\e'F-$ \"3S+3!3j&RpQF-7$$!3-LL3_n5/aF-$\"3[*GS'4tAMPF-7$$!3Tn;Hd*f)GUF-$\"3![ `TMF3Oi$F-7$$!3+M$ektb#RJF-$\"30krl'p(pNNF-7$$!3X+]7yO.6?F-$\"3m21I!o^ cX$F-7$$!3YRLe9\"[AW)!#>$\"3'[%>n?\"o>Q$F-7$$\"3-&**\\P%3U)=$Ffp$\"3>% *e]W5&eJ$F-7$$\"3\\LL$eM*>::F-$\"3;$e`&H[0aKF-7$$\"3Onm\"HZ^*oDF-$\"3_ *3+S)[)R?$F-7$$\"3;,+](yE_v$F-$\"31Dwr/Ow^JF-7$$\"3)z***\\iIPY\\F-$\"3 eF#Q_9HJ5$F-7$$\"3e****\\7KE%4'F-$\"3#\\^63<%QfIF-7$$\"3zk;aQulOrF-$\" 3\"fYL6g'*>-$F-7$$\"3%GLLe*\\;w$)F-$\"3;=l1)R/,)HF-7$$\"34KLL$[!>E%*F- $\"3mNIf)3ol%HF-7$$\"3!**\\(ou>wk5!#<$\"3KuSd#4v&4HF-7$$\"3)GLL3;zG<\" Fis$\"3#G%4&[4p%yGF-7$$\"3++vo\\3]\"H\"Fis$\"3wM$=7wXf%GF-7$$\"3=+D1\\ nX/9Fis$\"3m_;&*Q2S;GF-7$$\"3VL$3-Q9B_\"Fis$\"3G#Qhx^Gpy#F-7$$\"3ZL3xw WaI;Fis$\"3i(o!\\h`)4w#F-7$$\"3Sm;Hd_GZq$Gtq#F-7$$\"3++DJbS5u>Fis$\"3)*)*)yXb\")[o#F-7$$\"3q m;a=&4\")3#Fis$\"3kIQ&fQ%[hEF-7$$\"3Z****\\<$))e?#Fis$\"3_.R#*eZW]dUc&f #F-7$$\"3u**\\i?9QcDFis$\"3)y]1'*4[Md#F-7$$\"3%HLL3c2wm#Fis$\"3O;%Q6D8 Ub#F-7$$\"3s***\\i?ljy#Fis$\"30\\qs`FIMDF-7$$\"3SmT&QGxR*GFis$\"3HfHZ- $*y;DF-7$$\"3Y++v3di6IFis$\"3U*[4!=h=)\\#F-7$$\"3\"G$3-3PKAJFis$\"3!)H Bg*pt6[#F-7$$\"3++vV8].QKFis$\"35]D_kT(QY#F-7$$\"3)GL$ekR;^LFis$\"3yun 'eh9uW#F-7$$\"3I*\\(=#\\*fpMFis$\"34eAeE+kICF-7$$\"33nmm\")pm$e$Fis$\" 3\\%R'f\"y/\\T#F-7$$\"3dm;zkuJ+PFis$\"3[mDwP)>#*R#F-7$$\"3OL3-B?+;QFis $\"335EqQ@0%Q#F-7$$\"3T++] " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 11 "Example 11 " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " } {XPPEDIT 18 0 "Limit((1/(x+4)-1/4)/x,x=0)" "6#-%&LimitG6$*&,&*&\"\"\"F ),&%\"xGF)\"\"%F)!\"\"F)*&F)F)F,F-F-F)F+F-/F+\"\"!" }{TEXT -1 15 " if \+ it exists. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "Limit((1/(x+4)-1/4)/x,x=0):\neval_lim(%,info=true): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&*&\"\"\"F),&%\"xGF) \"\"%F)!\"\"F)#F)F,F-F)F+F-/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G-%&LimitG6$*&-F$6#,$*(\"\"%!\"\"%\"xG\"\"\",&F/F0F-F0F.F.F0F/F./F /\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*(-F$6#%\"x G\"\"\"F,!\"\",&*&\"\"%F-F,F-F-\"#;F-F.F./F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*&\"\"\"F*,&*&\"\"%F*%\"xGF*F*\"#;F*! \"\"F0/F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#!\"\"\"#;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "p1 := plot(-1/(4*x+16),x=-0.7..0.7):\np2 := plot([[[0,-1/16]]$2], style=point,\n symbol=[circle$2],symbolsize=[15,18],color=bro wn):\nt1 := plots[textplot]([[0.7,-.01,`x`],[-.03,0.04,`y`]],color=COL OR(RGB,.01,0,0)):\nplots[display]([p1,p2,t1],view=[-0.7..0.7,-0.11..0. 04],labels=[``,``],tickmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 523 193 193 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!3a**************p!#=$!3)f ddddddd(!#>7$$!3cmmmT)R[p'F*$!3#>oplKWj](F-7$$!3[KLe>;KHkF*$!3oX\")f@ \\(pW(F-7$$!3bmm;4'=28'F*$!3cq-[y)>8Q(F-7$$!3`mm;ki8IeF*$!3'R$)*3c&)Q; tF-7$$!3+LLeMD)4`&F*$!3awQjH-*GD(F-7$$!3Umm\"HtGOD&F*$!3w!pZV$f*\\>(F- 7$$!3i***\\i$\\Wm\\F*$!3)\\JO'Hd,OrF-7$$!3cmm\"H/R%pYF*$!3E)G+y!p-wqF- 7$$!3m***\\7l&QtVF*$!3wH$*oteA1pF-7$$!3#******\\ap')\\$F*$!3_D%HWTl!\\oF-7$$ !3)******\\noa>$F*$!3Yw:&o&=k#z'F-7$$!3f*****\\XyK!HF*$!3t)f;\"***R\"R nF-7$$!3$om;HuTzj#F*$!3c%\\'o#=!G\"p'F-7$$!3ULLLG2VABF*$!3$3fQ7dZ_j'F- 7$$!37LLL8::b?F*$!3=cN()Q'4&)e'F-7$$!3o++DYACW_!R'F-7$$!3%\\LL$eRj&z&F-$!3#zd!*e:))=M'F-7$$!3eHL$ e/'oSIF-$!39e$3*[Y(yH'F-7$$!3+ulmmTD5p!#@$!354bx8*z5D'F-7$$\"3_GL$3dPv ,$F-$!3'RG(>(*R?.iF-7$$\"3e!***\\ioY/dF-$!3e&*3\"H'47ihF-7$$\"3gKLL3TU 1')F-$!310IsgqN=hF-7$$\"31*******)HWg6F*$!3g;%oYu#ztgF-7$$\"3w)***\\n$ RPX\"F*$!3jXy:C!>3.'F-7$$\"3z***\\Pp=vt\"F*$!3Q.C!yi9)*)fF-7$$\"3U**** \\_sg_?F*$!3RV#eHgM\\%fF-7$$\"3zkmmO$GdL#F*$!3[=7HEx<0fF-7$$\"3&****** \\_?!QEF*$!3m$G->f6L'eF-7$$\"3.KL3x@%>\"HF*$!3@5[-iS)e#eF-7$$\"3x***** \\*3T6KF*$!3g>fVY)3by&F-7$$\"3_kmT?w=$\\$F*$!3]/Kp9m-[dF-7$$\"3-++v)[D xy$F*$!3a@0u[DO4dF-7$$\"33mmm\"4!pvSF*$!3/il3O41scF-7$$\"3Y)**\\PMirP% F*$!3()=\\T,$GNj&F-7$$\"3OMLL`f^nYF*$!3+\"3Bk]3pf&F-7$$\"3GKL$eXWW'\\F *$!3)3yb**f[*fbF-7$$\"3almTSU\"*e_F*$!3^YZ?'ptP_&F-7$$\"32+++!R,&HbF*$ !3#QjZLEX4\\&F-7$$\"3@nm;*zC'ReF*$!3;2%z3#oz`aF-7$$\"3fKLLtG+v%*z``F-7$$\"3a**************pF*$!3\"G@qh$*[\">`F--%'COLOURG6&%$ RGBG$\"#5!\"\"$\"\"!Fa[lF`[l-F$6&7#7$F`[l$!3+++++++]iF--%'SYMBOLG6$%'C IRCLEG\"#:-Fjz6&F\\[l$\")#)eqk!\")$\"))eqk\"Fa\\lFb\\l-%&STYLEG6#%&POI NTG-F$6&Fd[l-Fi[l6$F[\\l\"#=F]\\lFd\\l-%%TEXTG6%7$$\"\"(F_[l$F_[l!\"#Q \"x6\"-%&COLORG6&F\\[l$\"\"\"Fd]lF`[lF`[l-F^]l6%7$$!\"$Fd]l$\"\"%Fd]lQ \"yFf]lFg]l-%*AXESTICKSG6$\"\"&\"\"$-%+AXESLABELSG6%%!GF\\_l-%%FONTG6# %(DEFAULTG-%%VIEWG6$;$!\"(F_[lFa]l;$!#6Fd]lFa^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Exa mple 12 " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Limit(( 4-sqrt(x))/(x-16),x=16)" "6#-%&LimitG6$*&,&\"\"%\"\"\"-%%sqrtG6#%\"xG! \"\"F),&F-F)\"#;F.F./F-F0" }{TEXT -1 15 " if it exists. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "p1 : = plot((4-sqrt(x))/(x-16),x=-.3..17.7):\np2 := plot([[[16,-1/8]]$2],st yle=point,\n symbol=[circle$2],symbolsize=[15,18],color=brown ):\np3 := plot([[[0,-1/4]]$4],style=point,symbol=[circle$2,diamond,cro ss],color=red,\n symbolsize=[15,10$3]):\nt1 := plots[textp lot]([[17.7,-.05,`x`],[-.3,0.3,`y`]],color=COLOR(RGB,.01,0,0)):\nplots [display]([p1,p2,p3,t1],view=[-0.3..17.7,-0.51..0.3],labels=[``,``],ti ckmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 523 193 193 {PLOTDATA 2 "6.-%'CURVESG6$7X7$$\"33+++l/[Al!#?$!3'G>AgzA0X#!#=7$$\"3-+++ozLy=!#>$ !3NEDaxrM#F-7$$\"3T*******\\x[B*F 1$!3^Bz9u(zMK#F-7$$\"3&)*****fg!RIEF-$!38'GBG:$)e@#F-7$$\"3x*****>Y$HP VF-$!3%4qf([idY@F-7$$\"3E+++?]Zw\")F-$!3VxUc\"\\_!R?F-7$$\"3-+++=06/7! #<$!3#*Q(4\\M:='>F-7$$\"3'******4uO()e\"FZ$!3opQJ*ox4!>F-7$$\"3#****** f[M`%>FZ$!3[B0miJl`=F-7$$\"3$******fzqXJ#FZ$!3$RY@PDW6\"=F-7$$\"3=+++; bV'p#FZ$!3'y6b]:&Rs(FZ$!3urc&)zNH'\\\"F-7$$\"3?+++z68pvFZ$!3M*f4!op@\"[\"F- 7$$\"3!)*****>1Z[&zFZ$!3gN()3\"R#=m9F-7$$\"3H+++^Y04$)FZ$!3-'f?2J^HX\" F-7$$\"3/+++)Q:6p)FZ$!3t301W'[#R9F-7$$\"3-+++;\"pz3*FZ$!3#G+O9y#fD9F-7 $$\"3$3++SXJMV*FZ$!38^E+$H\\TT\"F-7$$\"3N+++S-a1)*FZ$!3Rpm]f>5!#;$!31C)Gj\"*Q.R\"F-7$$\"3/+++Z$4p0\"F]v$!3595V;u6z8F-7$ $\"3/+++h_R$4\"F]v$!3]`RtRdho8F-7$$\"31+++Zm!R8\"F]v$!3w$4gofRtN\"F-7$ $\"3)*******GzIq6F]v$!3i#*>:z:`Z8F-7$$\"3\"******z1u\"47F]v$!3:[&za\"z PP8F-7$$\"3(******fc#RW7F]v$!35M2**fbWG8F-7$$\"3++++'o&*GG\"F]v$!3cNuK Fh&*=8F-7$$\"3$******H7C\">8F]v$!3alXScoF58F-7$$\"3%******pF$*pN\"F]v$ !3yu6kH(\\9I\"F-7$$\"3)******zH4,Z\"F]v$!3!QUc5ZakF\"F-7$$ \"3(******fr&G3:F]v$!39AE43bWo7F-7$$\"3)******p6Yha\"F]v$!3i`E-n\"*pg7 F-7$$\"3-+++$*e$4e\"F]v$!3!ow3w!eu`7F-7$$\"3++++.*33i\"F]v$!3I&R.r(>'f C\"F-7$$\"37+++)zrkl\"F]v$!31*)*)e13;R7F-7$$\"35+++$)e\\%p\"F]v$!3va7M j%p?B\"F-7$$\"3%******Hy*)3t\"F]v$!3pN!\\LnIaA\"F-7$$\"3%************* p " 0 "" {MPLTEXT 1 0 403 "f := x -> (4-sqrt(x))/(x-16):\n'f(x)'=f(x);\nxvals := [15.9,15.99 ,15.999,16.,16.001,16.01,16.1]:\nyvals := [seq(evalf[15](f(xvals[i])), i=1..7)]: yvals := evalf[9](yvals):\nyvals := map(proc(_U) `if`(has(_U ,Float(undefined)),`undefined`,_U) end proc,yvals):\nA := array(1..2,1 ..9):\nA[1,1]:=x: A[2,1]:='f(x)': A[2,2]:=`|`: A[1,2]:=`|`:\nfor j to \+ 7 do A[1,j+2] := xvals[j]; A[2,j+2] := yvals[j] end do: evalm(A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&\"\"%\"\"\"*$F'#F+\" \"#!\"\"F+,&F'F+\"#;F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7$7+%\"xG%\"|grG$\"$f\"!\"\"$\"%*f\"!\"#$\"&**f\"!\"$$\"#;\"\"!$\"&, g\"F2$\"%,;F/$\"$h\"F,7+-%\"fG6#F(F)$!*Df>D\"!\"*$!*P&>]7FB$!*`>+D\"FB %*undefinedG$!*Z!)*\\7FB$!*v/)\\7FB$!*&H0[7FBQ(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Limi t((4-sqrt(x))/(x-16),x=16):\neval_lim(%,info=true):" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&LimitG6$*&,&\"\"%\"\"\"*$%\"xG#F)\"\"#!\"\"F),&F+F )\"#;F.F./F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,& \"\"%\"\"\"*$%\"xG#F+\"\"#!\"\"F+-F$6#,&F*F0F,F0F+,&F-F+\"#;F0F0F3F0/F -F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(-F$6#,&%\"xG\" \"\"\"#;!\"\"F-F+F/,&\"\"%F/*$F,#F-\"\"#F/F//F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&\"\"\"F),&\"\"%!\"\"*$%\"xG#F)\"\"#F,F ,/F.\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*&\"\"\"F *,&\"\"%F**$%\"xG#F*\"\"#F*!\"\"F1/F.\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#!\"\"\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }