Step-by-step Solution

Find the limit of $\frac{5x}{\tan\left(x\right)}$ as $x$ approaches $0$

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$5$

Step-by-step explanation

Problem to solve:

$\lim_{x\to0}\left(\frac{5x}{\tan\left(x\right)}\right)$
1

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5x\right)}{\frac{d}{dx}\left(\tan\left(x\right)\right)}\right)$
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The derivative of the linear function times a constant, is equal to the constant

$\lim_{x\to0}\left(\frac{5}{\frac{d}{dx}\left(\tan\left(x\right)\right)}\right)$

$5$
$\lim_{x\to0}\left(\frac{5x}{\tan\left(x\right)}\right)$

Limits

~ 0.33 seconds