{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 } {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 "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 83 "How long do you have to w ait to get within a certain percent of the limiting value?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 305 "This is the idea that underlies the defi nition of limits at infinity: if a limit exists you should be able to \+ get within a certain percentage of the limiting value by going out far enough towards infinity and then remain within that tolerance from th en on out, no matter how small a percentage you choose." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "C:=t->30*t/(200+t);\nLimit(C(t),t=i nfinity)=limit(C(t),t=infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "This approaches 30 from below as the graph readily shows, so if w e want to get within 1% of the asymptotic value 30, we need to cross t he line " }{XPPEDIT 18 0 "y = .99;" "6#/%\"yG-%&FloatG6$\"#**!\"#" } {TEXT -1 32 "*30; by trial and error we find:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([C(t),30,.99*30],t=0..40000,color=[red,blac k,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and narrowing our ve rtical window to see better:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([C(t),30,.99*30],t=0..40000,29..30,color=[red,black,blue]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "We can easily find the inters ection point where the graph crosses the 1% line:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 59 "T:=fsolve(C(t)=.99*30,t=0..40000);\n%/60*hr; \n%*days/(24*hr);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "but the tim e units of minutes are not very useful for interpretation so they shou ld be converted to a more reasonable unit." }}}}{MARK "9 0 0" 123 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }