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# 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.$\displaystyle \lim_{x\to 0} \frac{\ln (1 + x)}{\cos x + e^x - 1}$

## This limit can be evaluated by substituting 0 for $x, \quad \lim _{x \rightarrow 0} \frac{\ln (1+x)}{\cos x+e^{x}-1}=\frac{\ln 1}{1+1-1}=\frac{0}{1}=0$

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##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

Okay. So we see with this problem here we're going to be using reptiles role. It's the natural log of one plus X. And then that's going to be divided by the coastline of X plus me to the X. And this one. So with this in mind we realized that when we plug in zero, we don't actually get an indeterminate forms. So we won't have to use low tiles role in this case Plugging in zero. Here we get the natural log of one plus zero, which is just zero Plugging in zero. Here we get one, 1 -1 is zero. So that will go away. But then down here E to the zero is one. So the final answer is going to be 0/1 which is equal to zero. zero will be the final answer. So this problem that is the limit as X approaches zero of the given function.

California Baptist University

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Derivatives

Differentiation

Volume

##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

Lectures

Join Bootcamp