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However, the limit of " }}{PARA 0 "" 0 "" {TEXT -1 26 "the function as x goes to " }{XPPEDIT 18 0 "inf inity" "6#%)infinityG" }{TEXT -1 46 " is 0. The function is continuou s everywhere " }}{PARA 0 "" 0 "" {TEXT -1 17 "except at x = 0. " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 9 " Example 3" }}{PARA 0 "" 0 "" {TEXT -1 9 "a) Let " }{XPPEDIT 18 0 "s( x) = x^2 - 4*x" "6#/-%\"sG6#%\"xG,&*$F'\"\"#\"\"\"*&\"\"%F+F'F+!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t(x) = 4*x - x^2" "6#/-%\"tG6#%\"x G,&*&\"\"%\"\"\"F'F+F+*$F'\"\"#!\"\"" }{TEXT -1 35 ". Graph s(x) and \+ t(x) on the same " }}{PARA 0 "" 0 "" {TEXT -1 29 "axes with different \+ colors. " }}{PARA 0 "" 0 "" {TEXT -1 88 "b) Suppose that s(x) < f(x) < t(x) for all x. Find the limit of f(x) as x goes to 4. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:= x^2 - 4*x;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&*$)%\"xG\"\"#\"\"\"F**&\" \"%F*F(F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "t:= 4*x - x^2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG,&%\"xG \"\"%*$)F&\"\"#\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([s(x),t(x)], x= 2..6, color=[magenta, brown]);" }{TEXT -1 0 " " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$ 7S7$$\"\"#\"\"!$!\"%F*7$$\"+<')=(3#!\"*$!+X\")R#*RF07$$\"+m40j@F0$!+#Q 9M(RF07$$\"+7hO[AF0$!+uUJQRF07$$\"+#yYUL#F0$!+*3z#))QF07$$\"+w#>(>CF0$ !+Hd$Q#QF07$$\"+>K'*)\\#F0$!+1d.^PF07$$\"+Kd,\"e#F0$!+@2UiOF07$$\"+fX( em#F0$!+w5hcNF07$$\"+U7Y]FF0$!+Ez!oV$F07$$\"+V!pu$GF0$!++ck)H$F07$$\"+ ib59HF0$!+C5TkJF07$$\"+I,Q+IF0$!+'fR#**HF07$$\"+]*3q3$F0$!+V:T=GF07$$ \"+q=\\qJF0$!*y[*HE!\")7$$\"+fBIYKF0$!*VInW#F^p7$$\"+j$[kL$F0$!*x0R@#F ^p7$$\"+`Q\"GT$F0$!*,dR+#F^p7$$\"+s]k,NF0$!*3i]u\"F^p7$$\"+`dF!e$F0$!* aGF]\"F^p7$$\"+sgamOF0$!*>CEA\"F^p7$$\"+$[F0$\"*UI,-%F^p7$$\"+qfa<\\F0$\"*\\u?^%F^p7$ $\"+1O0)*\\F0$\"*VD$))\\F^p7$$\"+#G2A3&F0$\"*t,+]&F^p7$$\"+$)G[k^F0$\" *#>&R,'F^p7$$\"+7yh]_F0$\"*;;lc'F^p7$$\"+()fdL`F0$\"*')GF6(F^p7$$\"+-F T=aF0$\"*SXbo(F^p7$$\"+Epa-bF0$\"*(\\$yE)F^p7$$\"+Sv&)zbF0$\"*,!Q:))F^ p7$$\"+GUYocF0$\"*zHwX*F^p7$$\"+2^rZdF0$\"+_o`/5F^p7$$\"+sI@KeF0$\"+.d eo5F^p7$$\"+2%)38fF0$\"+)3E78\"F^p7$$\"\"'F*$\"#7F*-%'COLOURG6&%$RGBG$ \"*++++\"F^p$F*F*F][l-F$6$7S7$F($\"\"%F*7$F.$\"+X\")R#*RF07$F4$\"+#Q9M (RF07$F9$\"+uUJQRF07$F>$\"+*3z#))QF07$FC$\"+Hd$Q#QF07$FH$\"+1d.^PF07$F M$\"+@2UiOF07$FR$\"+w5hcNF07$FW$\"+Ez!oV$F07$Ffn$\"++ck)H$F07$F[o$\"+C 5TkJF07$F`o$\"+'fR#**HF07$Feo$\"+V:T=GF07$Fjo$\"*y[*HEF^p7$F`p$\"*VInW #F^p7$Fep$\"*x0R@#F^p7$Fjp$\"*,dR+#F^p7$F_q$\"*3i]u\"F^p7$Fdq$\"*aGF] \"F^p7$Fiq$\"*>CEA\"F^p7$F^r$\")%H1U*F^p7$Fcr$\")FQ\\jF^p7$Fhr$\")Sf*R $F^p7$F]s$\"'a$*yF^p7$Fbs$!)_%H_$F^p7$Fgs$!)5.&y'F^p7$F\\t$!*2dS/\"F^p 7$Fat$!*P\\hV\"F^p7$Fft$!*XNR$=F^p7$F[u$!*F#=KAF^p7$F`u$!*Fr(*o#F^p7$F eu$!*6hZ6$F^p7$Fju$!*muHe$F^p7$F_v$!*UI,-%F^p7$Fdv$!*\\u?^%F^p7$Fiv$!* VD$))\\F^p7$F^w$!*t,+]&F^p7$Fcw$!*#>&R,'F^p7$Fhw$!*;;lc'F^p7$F]x$!*')G F6(F^p7$Fbx$!*SXbo(F^p7$Fgx$!*(\\$yE)F^p7$F\\y$!*,!Q:))F^p7$Fay$!*zHwX *F^p7$Ffy$!+_o`/5F^p7$F[z$!+.deo5F^p7$F`z$!+)3E78\"F^p7$Fez$!#7F*-Fjz6 &F\\[l$\")#)eqkF^p$\"))eqk\"F^pFjdl-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEW G6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 55 "answer to (3b): limit of f(x) as x goes to 4 is zero. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 259 11 "Example 4 " }}{PARA 0 "" 0 "" {TEXT -1 8 "a) Let " }{XPPEDIT 18 0 "u(x) = (4x - 1)/ x" "6#/-%\"uG6# %\"xG*&,&*&\"\"%\"\"\"F'F,F,F,!\"\"F,F'F-" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v(x) = (4*x^2 + 3*x)/(x^2)" "6#/-%\"vG6#%\"xG*&,&*&\"\" %\"\"\"*$F'\"\"#F,F,*&\"\"$F,F'F,F,F,*$F'F.!\"\"" }{TEXT -1 25 ". Plo t u(x) and v(x) on " }}{PARA 0 "" 0 "" {TEXT -1 36 "the same axes with different color. " }}{PARA 0 "" 0 "" {TEXT -1 93 "b) Suppose that u( x) < f(x) < v(x) for all x > 5. Find limit of f(x) as x goes to infini ty. 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Plot m(x) to obtain the limit " }}{PARA 0 "" 0 "" {TEXT -1 82 "of m(x) as x goes to negative infinity. Compute the li mit without using Maple. 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