`lim_(x->0) (csc(x) - cot(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...

`lim_(x->0) (csc(x) - cot(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 50 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the limit, such that:

`lim_(x->0) (csc x - cot x) = lim_(x->0) (1 - cos x)/(sin x) = (1 - cos 0)/(sin 0) = (1-1)/0 = 0/0`

Since the limit is indeterminate `0/0` , you may use l'Hospital's rule:

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

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

Hence, evaluating the limit using l'Hospital's rule yields `lim_(x->0) (csc x - cot x) = 0.`

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