# What is L'hospital's rule used for?

Oct 7, 2014

L'hopital's rule is used primarily for finding the limit as $x \to a$ of a function of the form $f \frac{x}{g} \left(x\right)$, when the limits of f and g at a are such that $f \frac{a}{g} \left(a\right)$ results in an indeterminate form, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. In such cases, one can take the limit of the derivatives of those functions as $x \to a$. Thus, one would calculate ${\lim}_{x \to a} \frac{f ' \left(x\right)}{g ' \left(x\right)}$, which will be equal to the limit of the initial function.

As an example of a function where this may be useful, consider the function $\sin \frac{x}{x}$. In this case, $f \left(x\right) = \sin \left(x\right) , g \left(x\right) = x$. As $x \to 0$, $\sin \left(x\right) \to 0 \mathmr{and} x \to 0$. Thus,

lim_(x->0) sin(x)/x = 0/0 = ?

$\frac{0}{0}$ is an indeterminate form because we cannot precisely define what it is equal to.

However, by taking the derivatives, we find $f ' \left(x\right) = \cos \left(x\right) , g ' \left(x\right) = 1$. And thus...

${\lim}_{x \to 0} \sin \frac{x}{x} = {\lim}_{x \to 0} \cos \frac{x}{1} = {\lim}_{x \to 0} \cos \left(x\right) = \cos \left(0\right) = 1$