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Consider " }{XPPEDIT 18 0 "f(x) = x/(sqrt(1+3*x)-1);" "6#/-%\"fG6#%\"xG*&F'\"\"\" ,&-%%sqrtG6#,&F)F)*&\"\"$F)F'F)F)F)F)!\"\"F1" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "a) Use a graph \+ to find " }{XPPEDIT 18 0 "limit(f(x),x = 1);" "6#-%&limitG6$-%\"fG6#% \"xG/F)\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 "b) Use a table of values to find " } {XPPEDIT 18 0 "limit(f(x),x = 1)" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "c) Use the limit laws to find the exact value of " } {XPPEDIT 18 0 "limit(f(x),x = 1)" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"\" " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "2. Use the same function f as above, but this time cons ider what happens as x approaches zero." }}{PARA 0 "" 0 "" {TEXT -1 103 " Apply questions (a) - (c) of Problem 1 to this case. \+ Comment on your conclusions at each step." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 9 "Solution :" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }}{PARA 0 "" 0 "" {TEXT -1 98 "a ) We will define the function f and then we will plot it. When chosi ng the range of x in the " }{TEXT 275 4 "plot" }{TEXT -1 63 " comman d we keep in mind the fact that we are interested in the" }}{PARA 0 " " 0 "" {TEXT -1 34 " values of f for x near 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x->x /(sqrt(1+3*x)-1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG 6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&-%%sqrtG6#,&F.F.*&\"\"$F.F-F.F.F .F.!\"\"F6F(F(F(" }}}{PARA 261 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(x), x=0..2);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6%-%'CURVESG6$7fn7$$\"3#******\\8AB O\"!#?$\"3yx9#=LrMn'!#=7$$\"3#)******pUkCFF*$\"3Q(4\"Ri@E!o'F-7$$\"3s* ****\\Smp3%F*$\"3[.aTU#Rqo'F-7$$\"3k******R&)G\\aF*$\"3;Ql3bE!Qp'F-7$$ \"3Y******4G$R<)F*$\"3h(=up!))G2nF-7$$\"3%******zqd)*3\"!#>$\"3l)f#4n7 s?nF-7$$\"3*)*****>c'yM;FE$\"3)R3PWkFuu'F-7$$\"3%******pT:(z@FE$\"3g=$ f$Hn#Rx'F-7$$\"3')*****\\7t&pKFE$\"3saZ*f&=KEoF-7$$\"3')*****R$3VfVFE$ \"3=wg`\"eUz(oF-7$$\"35+++j&*)fD'FE$\"3!4:?#HR-mpF-7$$\"3K+++#H[D:)FE$ \"3gb)*3a&>?0(F-7$$\"3(******zbI=C\"F-$\"3h/ox<\"G&QsF-7$$\"3%******z! RBr;F-$\"3%p3O3Q!o)F-7$$\"3u *****p@Xt=%F-$\"3p#pW)Q9AS$)F-7$$\"35+++4y_qXF-$\"3)=B?.m)=m%)F-7$$\"3 i******\\1!>+&F-$\"3C:rJ/rR/')F-7$$\"3()******\\Z/NaF-$\"3aX*)o1#>'R() F-7$$\"3'*******\\$fC&eF-$\"3!3lqL-0o'))F-7$$\"3%******fz6:B'F-$\"3W'3 F+[?)z*)F-7$$\"3w*****p\"=C#o'F-$\"3#4z,XVI86*F-7$$\"3[+++n#pS1(F-$\"3 0S\\(4iQ/A*F-7$$\"3a*****HOD#3vF-$\"3YJwZJj'[M*F-7$$\"3>+++n(y8!zF-$\" 3][V\\Oa*GX*F-7$$\"3Z+++j.tK$)F-$\"3G+UFDyEp&*F-7$$\"3=+++)3zMu)F-$\"3 qLT1:#)4y'*F-7$$\"3G+++#H_?<*F-$\"3_4`u*)fp*y*F-7$$\"3)*******zihl&*F- $\"37HcU\")[]!*)*F-7$$\"3')******3#G,***F-$\"3+fLF&fJv***F-7$$\"34+++! o2J/\"!#<$\"3_U;y$>\"p55Fav7$$\"3#******RQ#\\\"3\"Fav$\"3E^MKj42?5Fav7 $$\"3\"******f\"*[H7\"Fav$\"3EautY&f+.\"Fav7$$\"3#*******pvxl6Fav$\"3a PJO@0BS5Fav7$$\"31+++`qn27Fav$\"3[;N$RWTAe\"Fav$\"3cRBqz3TK6Fav7$$\"3*)*****f!*3`i\"Fav$\" 3Aqc;'pV89\"Fav7$$\"3-+++$*zym;Fav$\"3)ec>&zY&)\\6Fav7$$\"3'******4N1# 4Xhu6Fav7$$\"3(******R6KU$=Fav$\"3)3#p/mhM$= \"Fav7$$\"37+++`v&Q(=Fav$\"3O^Z^a03\">\"Fav7$$\"3.+++Ol5;>Fav$\"3+I4@L 4D*>\"Fav7$$\"31+++/Uac>Fav$\"3tKchm%**p?\"Fav7$$\"\"#\"\"!$\"3-I:-P/D :7Fav-%'COLOURG6&%$RGBG$\"#5!\"\"$Fe]lFe]lF_^l-%+AXESLABELSG6$Q\"x6\"Q !Fd^l-%%VIEWG6$;F_^lFc]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 13 "" 1 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 163 "Observe that the values of y get clos er and closer to 1 as x is approaching 1 from the left ( x=0.4, \+ x=0.6, x=0.8, x=0.84, x=0.88, x=0.92, x=0.94, etc.) . " }}{PARA 0 "" 0 "" {TEXT -1 152 "Observe that the values of y get closer and close r to 1 as x is approaching 1 from the right (x=1.8, x=1.6, x=1.2 , x=1.16, x=1.12, x=1.08, etc)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 " Therefore, based on the graph, " } {XPPEDIT 18 0 "limit(f(x),x = 1)" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"\" " }{TEXT -1 4 " = 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 24 "b) Now in order to find " }{XPPEDIT 18 0 "limit(f(x),x \+ = 1)" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"\"" }{TEXT -1 133 " we will l ook at the values of f as x is getting closer and closer to 1. We should be careful to include values of x on both " }}{PARA 0 "" 0 " " {TEXT -1 143 " sides of 1. We will start with x approaching 1 from the left with first value x= .9 . Our x will increase toward \+ 1 in steps of .01." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "\nx[1]:=.9;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$\"\"*!\"\"" }}}{PARA 262 "" 1 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "\nfor nn from 2 to 10 do x[ nn]:=x[nn-1]+.01 od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\" \"#$\"#\"*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$$\"##* !\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$\"#$*!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&$\"#%*!\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$\"#&*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($\"#'*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")$\"#(*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"xG6#\"\"*$\"#)*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5 $\"#**!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 45 "We will create a table of \+ values of f for " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 9 " through " }{XPPEDIT 18 0 "x[10];" "6#&%\"xG6#\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "\nfor nn from 1 to 10 do f(x[nn]):=evalf(f(x[nn])) od ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"*!\"\"$\"+A!G^u*! #5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#\"*!\"#$\"+,$p5x*!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"##*!\"#$\"+wk!pz*!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#$*!\"#$\"+#ySE#)*!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#%*!\"#$\"+CMF[)*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#&*!\"#$\"+Cc!Q()*!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#'*!\"#$\"+Q&Q#**)*!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#(*!\"#$\"+9LdC**!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#)*!\"#$\"+#36)\\**!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#**!\"#$\"+WH&\\(**!#5" }} }{PARA 0 "" 0 "" {TEXT -1 70 "The values of f , as x approaches 1 from the left, approach 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "Let's examine lim f(x) as x comes close to 1 from the right." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "\nx[1]:=1.1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$\"#6!\"\"" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "\nfor nn from 2 to 10 do x[nn]:=x[nn-1]-.01 od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"xG6#\"\"#$\"$4\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\" \"$$\"$3\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$\"$2 \"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&$\"$1\"!\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$\"$0\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($\"$/\"!\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"xG6#\"\")$\"$.\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"*$\"$-\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5$\"$,\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "\nfor nn from 1 to 10 do f(x[nn]):=evalf(f(x[nn])) od;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"#6!\"\"$\"+X![X-\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$4\"!\"#$\"+6E8A5!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$3\"!\"#$\"+w'3(>5!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$2\"!\"#$\"+^hF<5!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$1\"!\"#$\"+V\\$[,\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$0\"!\"#$\"+f\\Q75!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$/\"!\"#$\"+/h#*45!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$.\"!\"#$\"+!Geu+\"!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$-\"!\"#$\"+*Q\")\\+\" !\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"$,\"!\"#$\"+I`\\-5 !\"*" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "In this case, too, the values of f seem to approach 1. Therefo re the numerical evaluation of f would indicate that " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "limit(f(x),x = 1)" "6#-%&limitG6$-%\"fG6#%\"xG/F) \"\"\"" }{TEXT -1 5 " = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "c) To obtain the limit algebraically in this case we can evaluate f(1)." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "limit(f(x),x = 1)" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"\"" }{TEXT -1 12 " = f(1) = 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 2 "2." }}{PARA 0 "" 0 "" {TEXT -1 94 " a) We will plot f(x ) with the range of x adjusted for the fact that x is approaching 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(f(x),x=-.5..0.5);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 285 176 176 {PLOTDATA 2 "6%-%'CURVESG6$7T7$$!3-+++n_qGL!#=$ \"3oJ<\\c$QvX$F*7$$!3(******H$p4ALF*$\"3L2W&zTlo_$F*7$$!3%)*****zf)[:L F*$\"3mM%=tdBsd$F*7$$!3F+++j-))3LF*$\"3Lw141L$)=OF*7$$!33+++%fjcH$F*$ \"3K(p/%zfo(o$F*7$$!3!******\\#pW#G$F*$\"3M!yPGZ%=XPF*7$$!3-+++'e8gD$F *$\"3Xn??Sp+TQF*7$$!3@+++[-eHKF*$\"3I3sAO!=9#RF*7$$!3'******4d8n<$F*$ \"3'>WYZfue0%F*7$$!3E+++%*o%Q7$F*$\"3?s$*H'pp*oTF*7$$!3!)*****H9(3:IF* $\"3#H/Wc![HjVF*7$$!3!******>RFj!HF*$\"3mRN(y(pPEXF*7$$!3*******f4OZr# F*$\"3i*eW-q'HpZF*7$$!3!******\\n\\!*\\#F*$\"3L%4>U0]4+&F*7$$!33+++DwZ #G#F*$\"3-E-/(>C\\?&F*7$$!3-+++D.xt?F*$\"3u2q?s!oBQ&F*7$$!37+++.TC%)=F *$\"3,nVb)4F6`&F*7$$!3!******>4z)e;F*$\"3G:^hmh&ep&F*7$$!33+++n`'zY\"F *$\"3uRJa4P!p#eF*7$$!3++++>t)eC\"F*$\"37s!HA_k6(fF*7$$!3%******ph5$\\5 F*$\"3'*G$\"3q$f-7h5*>iF*7$$!3'******* pXg#G'F^r$\"34IeVv-:OjF*7$$!3;+++]&Q(RTF^r$\"31\\Y&>>?GX'F*7$$!3%***** **4'=><#F^r$\"3!GHpYoTib'F*7$$!3I++++q*e$\\!#@$\"3y'3uQ!y>kmF*7$$\"33+ ++qRQb@F^r$\"3sxTa\")yusnF*7$$\"3!)******z\">Y2%F^r$\"3%>kLJTDX'oF*7$$ \"3++++!yXu9'F^r$\"3G/R_d0/hpF*7$$\"3%)*******[y))G)F^r$\"3%[ZQUoC\"eq F*7$$\"3.+++i_QQ5F*$\"3#[4?9-92:(F*7$$\"3%*******zZ3T7F*$\"3A))>@?)4#Q sF*7$$\"35+++O![hY\"F*$\"3s8ZQJs6LtF*7$$\"3-+++#Qx$o;F*$\"3`$fa[d9lT(F *7$$\"3++++u.I%)=F*$\"3u`&R.\\=P](F*7$$\"3-+++(pe*z?F*$\"3cIB$QT#>\"e( F*7$$\"3-+++C\\'QH#F*$\"3Q%e[#zjIkwF*7$$\"3#******H,M^\\#F*$\"3$RNdyHz 5u(F*7$$\"3%)*****\\?=bq#F*$\"3gU#y$Q]#*>yF*7$$\"3-+++2s?6HF*$\"3q$)=( 3.%p&*yF*7$$\"3********HXaEJF*$\"3+X?TZ7)F*7$$\"3E+++8tOcPF*$\"3Eo?8 #RVY>)F*7$$\"3?+++\\Qk\\RF*$\"3QKfoh8Yg#)F*7$$\"3w******o0;rTF*$\"33KV Xw5$[L)F*7$$\"3/+++lxGpVF*$\"3p/7?*eB/S)F*7$$\"37+++!oK0e%F*$\"3!ecp,F O%p%)F*7$$\"33+++<5s#y%F*$\"3COnEUEjM&)F*7$$\"3++++++++]F*$\"3pI1G+hz. ')F*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Ff[lFe[l-%+AXESLABELSG6$Q\"x6\"Q !F[\\l-%%VIEWG6$;$!\"&Fd[l$\"\"&Fd[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 29.000000 42.000000 0 0 "Curve 1" }}}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " The limit " } {XPPEDIT 18 0 "limit(f(x),x = 0);" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"! " }{TEXT -1 16 " is about 0.67" }}{PARA 0 "" 0 "" {TEXT -1 53 "b) As in Problem 1 we will create a table of values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "\nx[1]:=-0.1 ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$!\"\"F)" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "\nfor nn from 2 to 10 do x[nn]:=x[nn-1]+0.01 od;\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#$!\"*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$$!\")!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$!\"(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\" xG6#\"\"&$!\"'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$! \"&!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($!\"%!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")$!\"$!\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"*$!\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5$!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "\nfor nn from 1 to 10 do f(x[nn]):=evalf(f(x[nn])) od ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"\"F($\"+(3+A7'!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"*!\"#$\"+!eM8='!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\")!\"#$\"+ifERi!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"(!\"#$\"+/[1'H'!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"'!\"#$\"+X]z^j!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"&!\"#$\"+<[^1k!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"%!\"#$\"+xrFgk!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"$!\"#$\"+p188l!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"#F($\"+4*>^c'!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"\"!\"#$\"+VfG;m!#5" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Let's im prove on x to get more stabilized values of f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "\nx[1]:=-.00 1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$!\"\"!\"$" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "\nfor nn from 2 to 10 do x[nn]:=x[nn-1]+0.0001 od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#$!\"*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$$!\")!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$!\"(!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\" xG6#\"\"&$!\"'!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$! \"&!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($!\"%F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")$!\"$!\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"*$!\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5$!\"\"!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "\nfor nn from 1 to 10 do f(x[nn]):=evalf(f(x[nn])) od ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"\"!\"$$\"+qGmhm!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"*!\"%$\"+QN;im!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\")!\"%$\"+rSmim!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"(!\"%$\"+:];jm!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"'!\"%$\"+c]mjm!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"&!\"%$\"+$QlTm'!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"%F($\"+chmkm!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"$!\"%$\"+ki;lm!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>-%\"fG6#$!\"#!\"%$\"+-%\"fG6#$!\"\"!\"%$\"+#[khm'!#5" }}}{PARA 11 "" 1 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "By looking at the valu es above, we can state that the limit of f(x) as x approaches 0 \+ from the left is about 0.6666." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x[1]:=.001;\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$F'!\"$" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "for nn from \+ 2 to 10 do x[nn]:=x[nn-1]-0.0001 od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#$\"\"*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"xG6#\"\"$$\"\")!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\" %$\"\"(!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&$\"\"'!\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$\"\"&!\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($\"\"%!\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")$\"\"$!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"*$\"\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5$\"\"\"!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for nn from 1 to 10 do f(x[nn]):=evalf(f(x[nn])) od;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"\"!\"$$\"+<:mrm!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"*!\"%$\"+DN;rm!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\")!\"%$\"+1Nmqm!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"(!\"%$\"+;p;qm!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"'!\"%$\"+-%\"fG6#$\"\"&!\"%$\"+\"Ql\"pm!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"%!\"%$\"+nsmom!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"$!\"%$\"+*Hj\"om!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#$\"\"#!\"%$\"+-%\"fG6#$\"\"\"!\"%$\"+\"fbrm'!#5" }} }{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The lim it of f(x) as x approaches 0 from the right is about 0.6667. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Based on our observation of the values of f(x) as x approaches 0, we wi ll conjecture that " }{XPPEDIT 18 0 "limit(f(x),x = 0);" "6#-%&limitG 6$-%\"fG6#%\"xG/F)\"\"!" }{TEXT -1 9 " = 0.667." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "c) To obtain the limit a lgebraically, we cannot repeat what we did in Problem 1 as f(0) is no t defined. So " }{XPPEDIT 18 0 "limit(f(x),x = 0)" "6#-%&limitG6$-%\" fG6#%\"xG/F)\"\"!" }{TEXT -1 35 " , if it exists, cannot be equal to" }{TEXT 270 1 " " }{TEXT -1 179 " f(0). Based on the observations of \+ parts (a) and (b), we suspect that the limit does exist. Let's do so me algebraic manipulations on f(x). For all x in the domain of f \+ :" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x/(sqrt(1+3 *x)-1);" "6#/-%\"fG6#%\"xG*&F'\"\"\",&-%%sqrtG6#,&F)F)*&\"\"$F)F'F)F)F )F)!\"\"F1" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "x/(sqrt(1+3*x)-1);" "6 #*&%\"xG\"\"\",&-%%sqrtG6#,&F%F%*&\"\"$F%F$F%F%F%F%!\"\"F-" }{TEXT -1 2 " " }{XPPEDIT 18 0 "(sqrt(1+3*x)+1)/(sqrt(1+3*x)+1)" "6#*&,&-%%sqrt G6#,&\"\"\"F)*&\"\"$F)%\"xGF)F)F)F)F)F),&-F&6#,&F)F)*&F+F)F,F)F)F)F)F) !\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x(sqrt(1+3*x)+1)/(1+3*x-1);" "6#*&-%\"xG6#,&-%%sqrtG6#,&\"\"\"F,*&\"\"$F,F%F,F,F,F,F,F,,(F,F,*&F.F, F%F,F,F,!\"\"F1" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(sqrt(1+3*x)+1)/3; " "6#*&,&-%%sqrtG6#,&\"\"\"F)*&\"\"$F)%\"xGF)F)F)F)F)F)F+!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "So for x in the domain of f(x), " }{XPPEDIT 18 0 "li mit(f(x),x = 0);" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "limit((sqrt(3*x+1)+1)/3,x = 0);" "6#-%&limitG6$*&,& -%%sqrtG6#,&*&\"\"$\"\"\"%\"xGF.F.F.F.F.F.F.F.F-!\"\"/F/\"\"!" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "(sqrt(3(0)+1)+1)/3;" "6#*&,&-%%sqrtG6#,&- \"\"$6#\"\"!\"\"\"F-F-F-F-F-F-F*!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "limit(x/(sqrt(3*x+1)-1) ,x = 0);" "6#-%&limitG6$*&%\"xG\"\"\",&-%%sqrtG6#,&*&\"\"$F(F'F(F(F(F( F(F(!\"\"F0/F'\"\"!" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "2/3;" "6#*&\" \"#\"\"\"\"\"$!\"\"" }{TEXT -1 46 " . This confirms our guesses in ( a) and (b)." }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 147 " \+ \+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 262 10 "ASSIGNM ENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 0 "" }{TEXT 265 0 "" }{TEXT 266 10 "Problem 1." }}{PARA 0 "" 0 "" {TEXT -1 18 " Consider g(x) = \+ " }{XPPEDIT 18 0 "(sqrt(3+x)-sqrt(3))/x;" "6#*&,&-%%sqrtG6#,&\"\"$\"\" \"%\"xGF*F*-F&6#F)!\"\"F*F+F." }{TEXT -1 22 ". Find the " } {XPPEDIT 18 0 "limit(g(x),x = 0);" "6#-%&limitG6$-%\"gG6#%\"xG/F)\"\"! " }{TEXT -1 7 " by : " }}{PARA 0 "" 0 "" {TEXT -1 45 "a) Looking at t he graph of g(x) (use Maple)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "b) Creating a table of values of g(x) ( use Maple)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "c) Using algebraical computations." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 267 10 "Problem 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 0 "" }{TEXT 272 122 " Using Maple, try to determine the values of the following limits. Use graphing and function evaluation \+ at nearby points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "limit(sin(10*x)/x,x = 0);" "6#-%&limi tG6$*&-%$sinG6#*&\"#5\"\"\"%\"xGF,F,F-!\"\"/F-\"\"!" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "b) " } {XPPEDIT 18 0 "limit(h(x),x = 1);" "6#-%&limitG6$-%\"hG6#%\"xG/F)\"\" \"" }{TEXT -1 8 " where " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#/-%\"hG6#%\"xG-%*PIECEWISEG6%7$*&,&*$)F'\"\"%\"\"\"F1!\"\"F1F1,& F'F1F2F1F22F'F17$\"# " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "9 2" 10 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }