`lim_(x->1) (1 - x + ln(x))/(1 + cos(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...

`lim_(x->1) (1 - x + ln(x))/(1 + cos(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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Chapter 4, 4.4 - Problem 35 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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lemjay | High School Teacher | (Level 3) Senior Educator

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`lim_(x->1) (1-x+ln(x))/(1+cos(x))`

To compute for its limit, plug-in x=1.

`= (1-1+ln(1))/(1+cos(1))`

`=0/(1+cos(1))`

`=0`

The result is finite value. This means that the function is defined at x =1. Hence, there is no need to apply the L'Hospital's Rule in computing its limit as a x approaches 1.

Thus, `lim_(x->1) (1-x+ln(x))/(1+cos(x)) = 0` .    

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