{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {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 "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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 35 "Limit of a function at a \+ point (II)" }}{PARA 19 "" 0 "" {TEXT -1 19 "Wei-Chi Yang, Ph.D." }} {PARA 0 "" 0 "" {TEXT -1 11 "Exercise 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:=x->(2*x^3-x^2+3*x+1)/(x^3+2*x^2+4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,**& \"\"#\"\"\")9$\"\"$F0F0*$)F2F/F0!\"\"*&F3F0F2F0F0F0F0F0,(*$F1F0F0*&F/F 0F5F0F0\"\"%F0F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g :=proc(x,eps) abs(f(x)-2)-eps end;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%$epsG6\"F)F),&- %$absG6#,&-%\"fG6#9$\"\"\"\"\"#!\"\"F39%F5F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g(x,eps);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &-%$absG6#,&*&,**&\"\"#\"\"\")%\"xG\"\"$F,F,*$)F.F+F,!\"\"*&F/F,F.F,F, F,F,F,,(*$F-F,F,*&F+F,F1F,F,\"\"%F,F2F,F+F2F,%$epsGF2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve(g(x,10^(-2)),x,1..1000);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ln*R(\\!\"(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "evalf(f(497));evalf(f(498));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++?***)>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+#*>,!*>!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(\{f( x),1.98,2\},x=497..498,y=1.98..2.001);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"$(\\\"\"!$\"\"#F*7$$\"3y m;arz@q\\!#:F+7$$\"33$e9ui2/(\\F0F+7$$\"3km\"z_\"4iq\\F0F+7$$\"3wmT&ph N3(\\F0F+7$$\"3S$e*=)H\\5(\\F0F+7$$\"3c;z/3uCr\\F0F+7$$\"3!*\\7LRDXr\\ F0F+7$$\"3u;zR'ok;(\\F0F+7$$\"31]i5`h(=(\\F0F+7$$\"3YL$3En$4s\\F0F+7$$ \"3wmT!RE&Gs\\F0F+7$$\"3%)**\\K]4]s\\F0F+7$$\"3#)**\\PAvrs\\F0F+7$$\"3 %***\\nHi#H(\\F0F+7$$\"3'o\"z*ev:J(\\F0F+7$$\"3WL$347TL(\\F0F+7$$\"3IL LjM?`t\\F0F+7$$\"3#*\\7o7Tvt\\F0F+7$$\"3QLLQ*o]R(\\F0F+7$$\"3=]7=lj;u \\F0F+7$$\"3)*\\PaRY2a(\\F0F+7$$ \"3)o;zXu9c(\\F0F+7$$\"3A++&y))Ge(\\F0F+7$$\"3-+DE&QQg(\\F0F+7$$\"3!) \\7y%3Ti(\\F0F+7$$\"3E+v.[hYw\\F0F+7$$\"3cLLQx$om(\\F0F+7$$\"3%)**\\P+ V)o(\\F0F+7$$\"3g;zpe*zq(\\F0F+7$$\"3)***\\#\\'QHx\\F0F+7$$\"3I$e9S8& \\x\\F0F+7$$\"3u\\i?=bqx\\F0F+7$$\"31L$3s?6z(\\F0F+7$$\"3#*\\7`Wl7y\\F 0F+7$$\"3Wmm'*RRLy\\F0F+7$$\"3%o;a<.Y&y\\F0F+7$$\"39$e9tOc(y\\F0F+7$$ \"3!)***\\Qk\\*y\\F0F+7$$\"3?L3dg6!#<7$F.$\"3%\\L-nV#***)>Ffu7$F2 $\"3m)Rtl\"G***)>Ffu7$F5$\"3-)oVPC$***)>Ffu7$F8$\"3;4(3Pn$***)>Ffu7$F; $\"3y'\\$f,T***)>Ffu7$F>$\"3gyZE)\\%***)>Ffu7$FA$\"3+t<'*3\\***)>Ffu7$ FD$\"3[G(pOL&***)>Ffu7$FG$\"3'zvzpv&***)>Ffu7$FJ$\"3M#yoB>'***)>Ffu7$F M$\"3s#GGed'***)>Ffu7$FP$\"3s%oxu+(***)>Ffu7$FS$\"3O,?'3W(***)>Ffu7$FV $\"3]jDZey***)>Ffu7$FY$\"33_FnP#)***)>Ffu7$Ffn$\"3!G.T&)o)***)>Ffu7$Fi n$\"3a0QXq!****)>Ffu7$F\\o$\"3=bCn9&****)>Ffu7$F_o$\"3;1$[y!*****)>Ffu 7$Fbo$\"3!Q'*)=R.+!*>Ffu7$Feo$\"31(o#*)\\2+!*>Ffu7$Fho$\"3IuHQy6+!*>Ff u7$F[p$\"3%HZQ=d,+*>Ffu7$F^p$\"3Oq')>'*>+!*>Ffu7$Fap$\"397:&pV-+*>Ffu7 $Fdp$\"3!*>lf?G+!*>Ffu7$Fgp$\"3kn0\"\\B.+*>Ffu7$Fjp$\"3#fX,Hm.+*>Ffu7$ F]q$\"3s!er:3/+*>Ffu7$F`q$\"3%R2Bm[/+*>Ffu7$Fcq$\"3MS_KO\\+!*>Ffu7$Ffq $\"3#[7o.M0+*>Ffu7$Fiq$\"3+)3MFfu7$F\\r$\"3k2Oeih+!*>Ffu7$F_r$\" 3O\"R])*e1+*>Ffu7$Fbr$\"3Wy<%=*p+!*>Ffu7$Fer$\"3?PS+7u+!*>Ffu7$Fhr$\"3 )oGbF#y+!*>Ffu7$F[s$\"3_*fPFD3+*>Ffu7$F^s$\"3a)oEom3+*>Ffu7$Fas$\"3%[4 d-44+*>Ffu7$Fds$\"3%*Re95&4+*>Ffu7$Fgs$\"3Mu)[f*)4+*>Ffu7$Fjs$\"3+Xg3Q .,!*>Ffu7$F]t$\"3=Zc]L2,!*>Ffu7$F`t$\"3'p$42b6,!*>Ffu7$Fct$\"3;&QH&e:, !*>Ffu7$Fft$\"3QdU1#*>,!*>Ffu-Fit6&F[uF_uF\\uF_u-F$6$7S7$F($\"3)****** *******z>Ffu7$F.F]_l7$F2F]_l7$F5F]_l7$F8F]_l7$F;F]_l7$F>F]_l7$FAF]_l7$ FDF]_l7$FGF]_l7$FJF]_l7$FMF]_l7$FPF]_l7$FSF]_l7$FVF]_l7$FYF]_l7$FfnF]_ l7$FinF]_l7$F\\oF]_l7$F_oF]_l7$FboF]_l7$FeoF]_l7$FhoF]_l7$F[pF]_l7$F^p F]_l7$FapF]_l7$FdpF]_l7$FgpF]_l7$FjpF]_l7$F]qF]_l7$F`qF]_l7$FcqF]_l7$F fqF]_l7$FiqF]_l7$F\\rF]_l7$F_rF]_l7$FbrF]_l7$FerF]_l7$FhrF]_l7$F[sF]_l 7$F^sF]_l7$FasF]_l7$FdsF]_l7$FgsF]_l7$FjsF]_l7$F]tF]_l7$F`tF]_l7$FctF] _l7$FftF]_l-Fit6&F[uF\\uF\\uF_u-%+AXESLABELSG6$Q\"x6\"Q\"yFebl-%%VIEWG 6$;F(Fft;$\"$)>!\"#$\"%,?!\"$" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(g(x,10^(-2)),x=497.38..497.44, y=-0.000001..0.000001);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"3'***********zt\\!#:$\"379PU8!\\a$R! #C7$$\"3w*\\#HyI\"Q(\\F*$\"3M\"\\R8VjQn$F-7$$\"3+v[kdW#Q(\\F*$\"3Yux3E .EYMF-7$$\"3A]n\"\\DPQ(\\F*$\"37XEDq!)H!>$F-7$$\"3s\\s,P,&Q(\\F*$\"30a ')y_/kKHF-7$$\"3yu8*y&H'Q(\\F*$\"3OI6\"pv3in#F-7$$\"3CvG[W[(Q(\\F*$\"3 P+BSB\\YQCF-7$$\"3!\\()fB:()Q(\\F*$\"3/!)z#p9'H#>#F-7$$\"3?vQ=\"))**Q( \\F*$\"3F-7$$\"35vj=pD\"R(\\F*$\"3Y)y4^CSRo\"F-7$$\"3u*\\c.i DR(\\F*$\"3'[8#o**)3HU\"F-7$$\"3))\\U$e6PR(\\F*$\"3-sn'=[$*H>\"F-7$$\" 3q*\\>q0]R(\\F*$\"3%H+PARW;M*!#D7$$\"3')*\\U80jR(\\F*$\"3Y Fdo7$$\"35+XDn/+u\\F*$!3\\yx8lp#[S(!#E7$$\"3A+!y?#>,u\\F*$!3rbc!fDP9.$ Fdo7$$\"3Cv3wY_-u\\F*$!3%)=pSe)[jp&Fdo7$$\"33+IOTq.u\\F*$!3%4NS,IN_0)F do7$$\"3'[(3\">)*\\S(\\F*$!3e[xzx%HV1\"F-7$$\"3![isVIiS(\\F*$!39kG'pps 2J\"F-7$$\"3w\\(o:;vS(\\F*$!3OrO<\\!4zc\"F-7$$\"33v$)[op3u\\F*$!33')ps E$RS!=F-7$$\"37]i%Qq*4u\\F*$!3!p@knZO(e?F-7$$\"39vQIKH6u\\F*$!3TFF(Q'Q HBBF-7$$\"3oC^rZW7u\\F*$!3QQ&paq!f`DF-7$$\"3q\\Zn%)o8u\\F*$!3Yeyg;kJ-G F-7$$\"3s**4FL(\\T(\\F*$!3wGz33JFfIF-7$$\"3?]d6.B;u\\F*$!3)=N^l#Ql5LF- 7$$\"3wuo3lWKR#QF-7 $$\"3/+Ik-,?u\\F*$!3E<%oh5)emSF-7$$\"3/+D-eI@u\\F*$!3Bp\\_$ytcK%F-7$$ \"3%[(=_(zCU(\\F*$!3ib[B_FWgXF-7$$\"3;+b*=jPU(\\F*$!3$4hWt)f5<[F-7$$\" 3![(3/3(\\U(\\F*$!3i42y$)[ge]F-7$$\"3+vB4JBEu\\F*$!39+\"o-dR5J&F-7$$\" 3?+DVsYFu\\F*$!3=(Q2kQRyb&F-7$$\"3-v=n#f(Gu\\F*$!31nk17Y@;eF-7$$\"3?+! )RO+Iu\\F*$!3Y3ks0%f]1'F-7$$\"37]_!>w7V(\\F*$!3l%p_i'y`>jF-7$$\"3uu)Q? QDV(\\F*$!3uL&>c$y!>d'F-7$$\"33+5jypLu\\F*$!3:ss<.&3Q!oF-7$$\"3-]Ujp-N u\\F*$!3crLx=:fpqF-7$$\"3#)**fEd@Ou\\F*$!3A\"3t$>)4tI(F-7$$\"3)\\(3'>$ [Pu\\F*$!3ynG(*fewgvF-7$$\"31D6EjpQu\\F*$!3$=&eUJWN.yF-7$$\"3)******** ***Ru\\F*$!31SJJnw/k!)F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-%+A XESLABELSG6$Q\"x6\"Q\"yFg[l-%%VIEWG6$;$\"&Q(\\!\"#$\"&W(\\F_\\l;$F`[l! \"'$\"\"\"Fd\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(f(x), x=0..500,y=-5..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$\"\"!F)$\"3++++++++D!#=7$$\"3Xm;aQ`!e S$F,$\"3-$H*=nq^YYF,7$$\"3!HL$3x1h6oF,$\"3iG)4.48V7'F,7$$\"3#)*\\i:gT< -\"!#<$\"3>!)**=Cjf/sF,7$$\"3emmTN@Ki8F:$\"3KrYU$HjL4)F,7$$\"3l**\\7.K [V?F:$\"3'[+@'37S'e*F,7$$\"3;LL$3FWYs#F:$\"3379\"H<)4!3\"F:7$$\"3Bm;aQ `!eS$F:$\"3wo[u=kwy6F:7$$\"3I***\\iSmp3%F:$\"3It^X26Xf7F:7$$\"3!GLeRZF \"oZF:$\"3U%Rw#p\"GiK\"F:7$$\"3KmmmT&)G\\aF:$\"3_qC;\\\\A#Q\"F:7$$\"3e )***\\7G$R<)F:$\"3=vXc<*[q`\"F:7$$\"3GLLL3x&)*3\"!#;$\"3lrxY!*y4I;F:7$ $\"3!***\\i!R(*Rc\"Fbo$\"3dpu8z%egs\"F:7$$\"3`mm\"H2P\"Q?Fbo$\"3!pY%4t r^#y\"F:7$$\"3=LL$eRwX5$Fbo$\"3ouk8NxT^=F:7$$\"3ILL$3x%3yTFbo$\"3;*y)o u\"\\t)=F:7$$\"3Umm\"z%4\\Y_Fbo$\"3([C(>tL?4>F:7$$\"3QKLeR-/PiFbo$\"3S i.$z0XI#>F:7$$\"3u)**\\il'pisFbo$\"3v%)*pb\\ON$>F:7$$\"3LKLe*)>VB$)Fbo $\"3Rq&4;f\\<%>F:7$$\"3!*)**\\7`l2Q*Fbo$\"3%Q9oS%y8[>F:7$$\"3umm;/j$o/ \"!#:$\"3kFo]\\_R`>F:7$$\"3HLL3_>jU6Fer$\"33n?Mba@d>F:7$$\"3-++]i^Z]7F er$\"3%[@V].I3'>F:7$$\"3!*****\\(=h(e8Fer$\"3M3ua/I*Q'>F:7$$\"33++]P[6 j9Fer$\"31O)etDBk'>F:7$$\"3;L$e*[z(yb\"Fer$\"3)G5![>@Vo>F:7$$\"3[mm;a/ cq;Fer$\"3Qh9:w(G0(>F:7$$\"3mmmm;t,mF:7$$\"3'***\\iS j0x=Fer$\"3=Exf$fEP(>F:7$$\"3Smmm\"pW`(>Fer$\"3Q4N#***o,v>F:7$$\"3/+]i !f#=$3#Fer$\"3#**zS9A%Hw>F:7$$\"3'***\\(=xpe=#Fer$\"3)G0Imw%Rx>F:7$$\" 3gmm\"H28IH#Fer$\"3KBoap\"R%y>F:7$$\"3qm;zpSS\"R#Fer$\"3sw>\"Rr;$z>F:7 $$\"3ELL3_?`(\\#Fer$\"3#oD!z^l=!)>F:7$$\"3;L$e*)>pxg#Fer$\"3+()RF:7$$\"3a**\\Pf4t.FFer$\"3w^`p;Ko\")>F:7$$\"3LLLe*Gst!GFer$\"3m)*Qm> KN#)>F:7$$\"3#******\\#RW9HFer$\"3E$f*)oz&*H)>F:7$$\"3')***\\7j#>>IFer $\"3W1mI12e$)>F:7$$\"3g**\\i!RU07$Fer$\"37$*e.o&4T)>F:7$$\"3#)***\\(=S 2LKFer$\"35,#eCCeY)>F:7$$\"3Rmmm\"p)=MLFer$\"3!GY#p/*>^)>F:7$$\"38++]( =]@W$Fer$\"3[.@67Je&)>F:7$$\"3#GLe*[$z*RNFer$\"39c3KC'yf)>F:7$$\"3;++] iC$pk$Fer$\"3g-^a:pQ')>F:7$$\"3'fm\"H2qcZPFer$\"3Uy#fA&*\\n)>F:7$$\"3- +]7.\"fF&QFer$\"3A(3rKH4r)>F:7$$\"3Smm;/OgbRFer$\"3mDV\\eAW()>F:7$$\"3 g**\\ilAFjSFer$\"3?T(3u*Gx()>F:7$$\"3SLLL$)*pp;%Fer$\"3m5+L*Gv!))>F:7$ $\"3ILL3xe,tUFer$\"3_T;QT%p$))>F:7$$\"3em;HdO=yVFer$\"31bTvnrk))>F:7$$ \"3'******\\#>#[Z%Fer$\"3SfU%o#4*)))>F:7$$\"3emmT&G!e&e%Fer$\"3Kln%ztd \"*)>F:7$$\"3=LLL$)Qk%o%Fer$\"3/q'=vu&Q*)>F:7$$\"3k**\\iSjE!z%Fer$\"3) 41s:_='*)>F:7$$\"3))**\\P40O\"*[Fer$\"3oC'*>]>$)*)>F:7$$\"$+&F)$\"3))o )ott^+*>F:-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q\"yF a_l-%%VIEWG6$;F(Fb^l;$!\"&F)$\"\"&F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 225 "Remark: For this example, we note that if x > 497.3996765, then \+ the distance between f(x) and 2 is less than 10^(-2). By looking at th e graph of f above, we notice that f has an horizontal asymoptote at \+ y =2 (from below). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 12 "Exercise 2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->x/(x-3)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"x G6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F-F.\"\"$!\"\"!\"#F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g:=proc(x,eps) abs(f(x))-eps end;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%$epsG6\"F)F),&-%$absG6#-%\"fG6#9$\"\"\" 9%!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g(x,eps);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$absG6#*&%\"xG\"\"\",&F(F)\"\"$! \"\"!\"#F)%$epsGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolv e(g(x,10^(6)),x,1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\%o#)*H !\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(g(x,10^(6)), x,3..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^Dt,I!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(f(x),x=1..4,y=-100..100);" }} {PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6%-%'CURVESG6$7ao7 $$\"\"\"\"\"!$\"3++++++++D!#=7$$\"31++]i9Rl5!#<$\"3mWm()*\\xl%GF-7$$\" 34+]PC#)GA6F1$\"3=X?7;E2$=$F-7$$\"3%****\\Peui=\"F1$\"3fl!HWQOhg$F-7$$ \"33++D'3&o]7F1$\"3'4h\"3P!pq3%F-7$$\"3/+](oX*y98F1$\"3kr8r%*pkHYF-7$$ \"3#***\\P9CAu8F1$\"3iZC0O*y\"*>&F-7$$\"3.+]P*zhdV\"F1$\"3-o:dzx\"y'eF -7$$\"3++]P>fS*\\\"F1$\"3E)4)Q<3vemF-7$$\"35+](=$f%Gc\"F1$\"3C$[>6;_nc (F-7$$\"3$*****\\#y,\"G;F1$\"3[1#oSVJ/l)F-7$$\"37++Dr\"zbo\"F1$\"3gYkc M4>c(*F-7$$\"3%*****\\(4&G]F1 $\"3R=@W1\\*[q\"F17$$\"3,++]siL-?F1$\"3N*HT&o^s6?F17$$\"3-+++!R5'f?F1$ \"3@6e$[o***GBF17$$\"3)***\\P/QBE@F1$\"3'=0!GG,(\\y#F17$$\"3!******\\ \"o?&=#F1$\"3M:'=BkH:H$F17$$\"3%)**\\Pa&4*\\AF1$\"33;gqnY())*RF17$$\"3 &)**\\7j=_6BF1$\"39$p%\\Njgw[F17$$\"33++vVy!eP#F1$\"3qxI[q8#y4'F17$$\" 3K+](=WU[V#F1$\"39Lav@H5BwF17$$\"3)****\\7B>&)\\#F1$\"3AHZb)=V^$**F17$ $\"3)***\\P>:mkDF1$\"31A;\\tMC`8!#;7$$\"3'***\\iv&QAi#F1$\"3#=uTfySv$= Ffs7$$\"31++vtLU%o#F1$\"3IS)HE_7bp#Ffs7$$\"37+++bjm[FF1$\"3KdJPPuJ^VFf s7$$\"3\"****\\(yb^6GF1$\"3@o]!e0#)Q\"zFfs7$$\"3)***\\PMaKsGF1$\"3%)oB +Lp2i*es=-R\"!#87$$\"3wuoH9y8*)HF1$\"3'>e*en]WLDF`w7$$\"31]7`9'HH* HF1$\"3O)))e=k9q)fF`w7$$\"3EP%[Y^D[*HF1$\"3eXUPAJ]=6!#77$$\"3#\\ilZT@n *HF1$\"3'4:$4\\d(yy#F`x7$$\"3e7G)[J<')*HF1$\"3sTlD&oj%o:!#67$$\"3z**** *\\@80+$F1$\"3\"\\lj#H=>R6!#57$$\"3nuo/$\\PD+$F1$\"332VxCG9jYF`x7$$\"3 c\\P4ra?23\"F`w7$$\"3s*\\i:b/[-$F1$\"3ND6fwnD;\\F `v7$$\"3q***\\Pm,H.$F1$\"3s9s!>K1(F`u7$$\"3s*\\P4')QY4$F1$\"3 kBfZ'H#=bMF`u7$$\"3()**\\P4w)R7$F1$\"3%ziFV\"e8K?F`u7$$\"3;++]x%f\")=$ F1$\"3OI8=+b30!*Ffs7$$\"3!)**\\P/-a[KF1$\"3sbLOx.!*e_Ffs7$$\"3/+](=Yb; J$F1$\"3f!)fF***Q&4MFfs7$$\"3')****\\i@OtLF1$\"3j`IkVv#*>CFfs7$$\"3')* *\\PfL'zV$F1$\"3A3`.K=O#z\"Ffs7$$\"3>+++!*>=+NF1$\"3Oy%)=bT0*R\"Ffs7$$ \"3-++DE&4Qc$F1$\"3liHe9P6@6Ffs7$$\"3=+]P%>5pi$F1$\"3O'3g$4URG#*F17$$ \"39+++bJ*[o$F1$\"3aN&eD&zfbyF17$$\"33++Dr\"[8v$F1$\"3'\\#RD.=:XmF17$$ \"3++++Ijy5QF1$\"37s:vz%zpz&F17$$\"31+]P/)fT(QF1$\"3I#yTF\")e)p]F17$$ \"31+]i0j\"[$RF1$\"3$\\3=J^)o-XF17$$\"\"%F*F`al-%'COLOURG6&%$RGBG$\"#5 !\"\"$F*F*Fial-%+AXESLABELSG6$Q\"x6\"Q\"yF^bl-%%VIEWG6$;F(F`al;$!$+\"F *$\"$+\"F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Exercise 3. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=x->(x^3-x^2-27*x+29)/(x^2 +3*x-10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(*&,**$)9$\"\"$\"\"\"F2*$)F0\"\"#F2!\"\"*&\"#FF2F0F2F6 \"#HF2F2,(F3F2*&F1F2F0F2F2\"#5F6F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(f(x),x=-50..50,y=-100..100);" }}{PARA 13 "" 1 " " {GLPLOT2D 384 384 384 {PLOTDATA 2 "6%-%'CURVESG6$7^p7$$!#]\"\"!$!3# \\K'yCjy*Q&!#;7$$!3GLLLe%G?y%F-$!3+1;6JhLr^F-7$$!3!pm;aesBf%F-$!33IC`( \\D7)\\F-7$$!3ALL$3s%3zVF-$!35EQzPtPnZF-7$$!3MLL$e/$QkTF-$!3E7>Nx70_XF -7$$!3'om;/\"=q]RF-$!3_3Dh%>ywL%F-7$$!3QLL3_>f_PF-$!3QGVg`I&)QTF-7$$!3 )****\\(o1YZNF-$!31W5y1C*G$RF-7$$!3DLL3-OJNLF-$!36&G8&)*Qx>PF-7$$!3()* **\\P*o%Q7$F-$!3&f'oG6))>2NF-7$$!3Kmmm\"RFj!HF-$!3cUsgj#e$)G$F-7$$!3+L L$e4OZr#F-$!3;Ys_ZJT&4$F-7$$!3[*****\\n\\!*\\#F-$!39tb#y*)Hz(GF-7$$!3% ******\\ixCG#F-$!3y&oOnes\"fEF-7$$!3c*****\\KqP2#F-$!3'>H)eX!oyW#F-7$$ !31LL3-TC%)=F-$!3[zm#e2-aD#F-7$$!3/mmm\"4z)e;F-$!3T_2zxA[D?F-7$$!3ommm m`'zY\"F-$!3ia[V&[<$H=F-7$$!3m***\\(=t)eC\"F-$!3E)3@B6D$)f\"F-7$$!3Omm m;1J\\5F-$!3!)4Ud+\"))))Q\"F-7$$!3S$***\\(=[jL)!#<$!3V/d+Z_mW6F-7$$!3f &****\\Pw%4tFjq$!3GQ(yDpA@,\"F-7$$!3y(***\\iXg#G'Fjq$!3A=4e)fs5O)Fjq7$ $!3?j;Hd!))ou&Fjq$!3!)*)\\2K$e=o'Fjq7$$!3hGL3_:<6_Fjq$\"3MBsn:HAen!#=7 $$!3y'GK*))p?W^Fjq$\"38Z7a*)pmW^Fjq7$$!3'\\C\"yDCCx]Fjq$\"3q#*H*Qy:Rs \"F-7$$!39.-jiyF5]Fjq$\"37.o$G`1,'=!#:7$$!3Jh\"z%*H8L%\\Fjq$!3]\"ofXr) GzVF-7$$!3[>\"Gjt[j([Fjq$!338F[+#f9Y#F-7$$!3nxqCu%Fjq$!3l2*zB_)Q9 F-7$$!3tE3F%z'f2WFjq$!3(oui-:[:8\"F-7$$!3WfmmT&Q(RTFjq$!3)3!oBqH*f(**F jq7$$!3Ihm\"HdGe:$Fjq$!3iHa!Grg%ewFjq7$$!3;jm;/'=><#Fjq$!37Q=Z?#=+;'Fj q7$$!3/'**\\7yQ16\"Fjq$!31kKB0*H/m%Fjq7$$!3%G*GLLe*e$\\!#>$!37:Dv_YZ*) HFjq7$$\"3_7nTgxE=]Fas$!3cs\"*yx,Df=Fjq7$$\"3WrmT5D,`5Fjq$!3#H>X_**4Y4 \"Fas7$$\"3_rTgx`gG8Fjq$\"37H9s>5*3[\"Fjq7$$\"3hr;zW#)>/;Fjq$\"3S)fiKh 74)[Fjq7$$\"3iY&)ehk4t;Fjq$\"3-gV0vXS]lFjq7$$\"3m@aQyY*>u\"Fjq$\"3)=1X \\q8I2*Fjq7$$\"3o'H#=&*G*3\"=Fjq$\"3PJsX\\r7Q8F-7$$\"3qr\"z>6\"zz=Fjq$ \"3U+bd*=kXD#F-7$$\"3sYgxG$*o[>Fjq$\"3:=%yw_BGh&F-7$$\"3v@HdXveN3+KN)o$F-7$$\"3!=nm\"zRQb@Fjq$! 3;o1OdN6V@F-7$$\"370vozhLvAFjq$!3AE1[MO`*G\"F-7$$\"3WQ$3-Q)G&R#Fjq$!3! \\H&p^*\\XY*Fjq7$$\"3yr\"H2eS_^#Fjq$!3OBvoT!3Md(Fjq7$$\"360+D\"y#>NEFj q$!3ih6XKxs\\jFjq7$$\"3wr;H#=(4vGFjq$!3$o4MDheq![Fjq7$$\"3VQLL$e,]6$Fj q$!3VwOe&oM?#QFjq7$$\"3urmT&Q5[f$Fjq$!3F))>+W&*)*=DFjq7$$\"310+](=>Y2% Fjq$!3My*)4`U#=f\"Fjq7$$\"3APLe*[K56&Fjq$!3/)*3i&)H83^Fgw7$$\"3Qpmm\"z Xu9'Fjq$\"3rk]>NdpW7Fjq7$$\"3q-+++&y))G)Fjq$\"3I?+&*>TMhOFjq7$$\"3I++] i_QQ5F-$\"3a'o()=Y:g*eFjq7$$\"3>++D\"y%3T7F-$\"3ID0+Hf\"y+)Fjq7$$\"3m* ***\\P![hY\"F-$\"3*f\"\\h^>GK5F-7$$\"3MLLL$Qx$o;F-$\"3.g!)\\\"=B(Q7F-7 $$\"3%3++]P+V)=F-$\"3!fhT.i+\"e9F-7$$\"3Amm\"zpe*z?F-$\"3%yPW%Q)[il\"F -7$$\"3;,++D\\'QH#F-$\"3eIJ,6)yB(=F-7$$\"3yKLe9S8&\\#F-$\"3-!*4!\\Q&Qv ?F-7$$\"3!4+]i?=bq#F-$\"3w=m\\%R0tG#F-7$$\"34LLL3s?6HF-$\"3Umcw***yU\\ #F-7$$\"3h++DJXaEJF-$\"3rBDPU'y2r#F-7$$\"3&zmmm'*RRL$F-$\"3R2%Hhy]\">H F-7$$\"3)ommTvJga$F-$\"3\"z3\\[zA@8$F-7$$\"3'[L$e9tOcPF-$\"3?E(**y*GBV LF-7$$\"33,++]Qk\\RF-$\"36ZyQ[$[r`$F-7$$\"3uLL$3dg6<%F-$\"3AJd+8XKfPF- 7$$\"3]nmmmxGpVF-$\"396!*>;\\)z&RF-7$$\"3K,+D\"oK0e%F-$\"35oH)RgZ(pTF- 7$$\"3Y,+v=5s#y%F-$\"3)Gw5>w)QsVF-7$$\"#]F*$\"37OOOOO6!f%F--%'COLOURG6 &%$RGBG$\"#5!\"\"$F*F*Fhel-%+AXESLABELSG6$Q\"x6\"Q\"yF]fl-%%VIEWG6$;F( F]el;$!$+\"F*$\"$+\"F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " g:=proc(x,eps) abs(f(x))-eps end;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%$epsG6\"F)F),&-%$ absG6#-%\"fG6#9$\"\"\"9%!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g(x,eps);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$absG 6#*&,**$)%\"xG\"\"$\"\"\"F-*$)F+\"\"#F-!\"\"*&\"#FF-F+F-F1\"#HF-F-,(F. F-*&F,F-F+F-F-\"#5F1F1F-%$epsGF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve(g(x,10^(3)),x,3..10000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+w\\+/5!\"'" }}}}{MARK "22" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }