{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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "L'Hospital Rule" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=x-> sin(x)^tan(x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-1..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f(x),x=0,right);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now, we ask Maple to compute the l imit step by step." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ln(f(x ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=x->tan(x)*ln(sin( x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g1:=x->ln(sin(x))/cot(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "Since numerator goes to -infinity and denomina tor goes to infinity when x approaches to 0 from the right, we can app ly the L'Hospital Rule." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " num:=x->ln(sin(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(num );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "den:=x->cot(x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(den);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify(csc(x)^2-(1+cot(x)^2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "We simplify D(num)/D(den) by hand and yi eld -cos(x)*sin(x), and we see this goes to 0 when x approaches to 0 f rom the right. This convinces us that the limit of " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit(exp(ln(f(x))),x=0,right);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "18" 0 } {VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }