L'Hopital's Rule

Some rich guy (L’Hopital) paid a poorer Mathematician (Bernoulli) so that the rule could be named after him.

The rule is straight forward:

Rule

If
$li{m}_{x\to a}\frac{f(x)}{g(x)}=\frac{0}{0}$ AND

Then
$li{m}_{x\to a}\frac{f(x)}{g(x)}=li{m}_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$

This comes from the idea that derivatives contain the idea of tiny changes. If we make a tiny change around $f(x)$ or $g(x)$ (I.E. Making $f^{\prime }(x)$ or $g^{\prime }(x)$), we can solve those types of limits.