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{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);" }}}
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