L'H�pital's Rule

Discovered By

Guillaume de l'Hopital

The Rule

For $\frac {0}{0}$ or $\frac {\infty}{\infty}$ case, the limit $\lim_{x \to a} \cfrac {f(x)}{g(x)} = \lim_{x \to a} \cfrac {f'(x)}{g'(x)}$ where $f'(x)$ and $g'(x)$ are the first derivatives of $f(x)$ and $g(x)$ , respectively.

Examples

$\lim_{x \to 4} \cfrac {x^3 - 64}{4 - x} = \lim_{x \to 4} \cfrac {3x^2}{-1} = -3(4)^2 = -3(16) = \boxed {-48}$

$\lim_{x \to 0} \cfrac {\sin x}{x} = \lim_{x \to 0} \cfrac {\cos x}{1} = \cos (0) = \boxed {1}$

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L'H�pital's Rule