L’Hopital’s Rule

Theorem: Consider a limit of the form ${\lim }_{x\to a}\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are differentiable on some open interval containing $a$ (except possible at $a$ itself). If ${\lim }_{x\to a}f(x)=0$ and ${\lim }_{x\to a}g(x)=0$ (so that the limit is of indeterminate form $\frac{0}{0}$), then ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ = ${\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$, if this limit exists.

The same result holds for the indeterminate form $\frac{\infty }{\infty }$

Warning

Make sure it is in one of the two forms ($\frac{\infty }{\infty }$ or $\frac{0}{0}$) only before applying L’Hopital’s rule, else it won’t work.