L'Hopital's Rule

For $\frac00$ or $\frac{\infty}{\infty}$ forms, $\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}$ .

L'Hopital's Rule says that if $\lim_{x\to a}\frac{f(x)}{g(x)}$ has the indeterminate form $\frac00$ or $\frac{\infty}{\infty}$ , then $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$ , provided the second limit exists (or is $\pm\infty$ ). Differentiating the numerator and denominator SEPARATELY (not applying the quotient rule) often turns an indeterminate limit into one that's trivial to evaluate directly — and can be applied repeatedly if the result is still indeterminate.

Examples

$\lim_{x\to0}\frac{\sin x}{x}\overset{\frac00}{=}\lim_{x\to0}\frac{\cos x}{1}=1$ .