`lim_(x->0) sin(4x)/tan(5x)` 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->0) sin(4x)/tan(5x)` 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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Chapter 4, 4.4 - Problem 12 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101 | Student, Graduate | (Level 2) Adjunct Educator

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Given the limit function `lim_{x->0}sin(4x)/tan(5x)` . We have to find the limit.

Applying the limit we can see that,

`lim_{x->0}sin(4x)/tan(5x)=0/0`

which is of the form `0/0` . Hence we have to apply L'Hospital's rule.

i.e. differentiating both numerator and denominator and then applying the limits we get,

`lim_{x->0}(4cos(4x))/(5sec^2(5x))=4/5`

hence the limit is `4/5`

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