`lim_(x->0) (cot(x) - 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->0) (cot(x) - 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 52 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`lim_(x->0)cot(x)-1/x`

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

`=lim_(x->0)(xcos(x)-sin(x))/(xsin(x))`

Apply L'Hospital rule , Test condition:0/0

`=lim_(x->0)((xcos(x)-sin(x))')/((xsin(x))')`

`=lim_(x->0)(x(-sin(x))+cos(x)-cos(x))/(xcos(x)+sin(x))`

`=lim_(x->0)(-xsin(x))/(sin(x)+xcos(x))`

Apply L'Hospital rule , Test condition:0/0

`=lim_(x->0)(-xcos(x)-sin(x))/(cos(x)-xsin(x)+cos(x))`

`=lim_(x->0)(-sin(x)-xcos(x))/(2cos(x)-xsin(x))`

`=lim_(x->0)(sin(x)+xcos(x))/(xsin(x)-2cos(x))`

`=(sin(0)+0cos(0))/(0sin(0)-2cos(0))`

`=0`

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