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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.$\displaystyle \lim_{x\to 0} \frac{\tanh x}{\tan x}$

## This limit has the form $\frac{0}{0} . \lim _{x \rightarrow 0} \frac{\tanh x}{\tan x}=\lim _{x \rightarrow 0} \frac{\operatorname{sech}^{2} x}{\sec ^{2} x}=\frac{\operatorname{sech}^{2} 0}{\sec ^{2} 0}=\frac{1}{1}=1$

Derivatives

Differentiation

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

So for this problem we're wanting to use little petals role. Um and we're given tangent Hx over kanji max. This is the graph. We see that when we plug in zero though we get 0/0, which is an indeterminant form. So we're gonna want to apply look towels rule. And we see that when we take the derivative of the numerator, we get second H. X squared. Um let's actually put the squared and here And then the nominator is just going to give us 2nd squared of X. So based on this, we can now plug in zero. And we see that Seeking. H0 is going to give us one and then seeking squared of X is going to give us one. So 1/1, which is one will be our final answer for a woman.

California Baptist University

#### Topics

Derivatives

Differentiation

Volume

##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

Lectures

Join Bootcamp