`lim_(x->oo) xtan(1/x)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply,...

`lim_(x->oo) xtan(1/x)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

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Textbook Question

Chapter 4, 4.4 - Problem 46 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`lim_(x->oo)xtan(1/x)`

`=lim_(x->oo)tan(1/x)/(1/x)`

Apply L'Hospital's Rule,

Test condition:0/0

`=lim_(x->oo)(tan(1/x)')/((1/x)')`

`=lim_(x->oo)(sec^2(1/x)(-1x^-2))/(-1x^-2)`

`=lim_(x->oo)sec^2(1/x)`

`=lim_(x->oo)1/(cos^2(1/x))`

`lim_(x->oo)cos(1/x)=cos(0)=1`

`=1/1`

=1

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