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L’Hôpital’s Rule

SUMMARY: L'Hôpital's Rule
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L’Hôpital’s Rule:

$$\text{If } \displaystyle{\lim_{x \to a} \dfrac{f(x)}{g(x)} \to \dfrac{0}{0}} \text{ or } \dfrac{\infty}{\infty}\text{,}$$

$$\text{then: }\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{\frac{d}{dx}\left[f(x)\right]}{\frac{d}{dx}\left[g(x)\right]}$$

That is, if you take a limit and it’s in the form  $\dfrac{0}{0}$  or  $\dfrac{\infty}{\infty}$,  then you can take the derivative of the numerator and the derivative of the denominator and find that limit instead.

Warning: Before taking the derivatives, verify that the original limit is in the form  $\dfrac{0}{0}$  or  $\dfrac{\infty}{\infty}$. Otherwise you cannot use L’Hôpital’s Rule.

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