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Step-by-step Solution

Find the limit of $\frac{1-e^x}{\tan\left(x\right)}$ as $x$ approaches $0$

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Answer

$-1$

Step-by-step explanation

Problem to solve:

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

As the limit results in indeterminate form, we can apply L'Hôpital's rule

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

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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Answer

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

Main topic:

Limits

Related formulas:

5. See formulas

Time to solve it:

~ 1.43 seconds