# Step-by-step Solution

## Find the limit of (1-e^x)/(tan(x) as x approaches 0

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## Step-by-step explanation

Problem to solve:

$\lim_{x\to0}\left(\frac{\left(1-e^x\right)}{\tan\left(x\right)}\right)$
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As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(1-e^x\right)}{\frac{d}{dx}\left(\tan\left(x\right)\right)}\right)$
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The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(1-e^x\right)}{\sec\left(x\right)^2\frac{d}{dx}\left(x\right)}\right)$

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$\lim_{x\to0}\left(\frac{\left(1-e^x\right)}{\tan\left(x\right)}\right)$

### Main topic:

Limits by L'Hôpital's rule

~ 0.32 seconds