L'Hôpital's rule states that given an unresolvable limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ the ratio of functions $\frac{f(x)}{g(x)}$ will have the same limit at $c$ as the ratio $\frac{f'(x)}{g'(x)}$ In short, if the limit of a ratio of functions approaches an indeterminate form, then $$\lim_{x\rightarrow c}\frac{f(x)}{g(x)} = \lim_{x\rightarrow c}\frac{f'(x)}{g'(x)}$$ provided this last limit exists. L'Hôpital's rule may be applied indefinitely as long as the conditions are satisfied. However it is important to note, that the nonexistance of $\lim\frac{f'(x)}{g'(x)}$ does not prove the nonexistance of $\lim\frac{f(x)}{g(x)}$

Example: We try to determine the value of $$\lim_{x\to \infty}\frac{x^2}{e^x}.$$ As $x$ approaches $\infty$ the expression becomes an indeterminate form $\frac{\infty}{\infty}$ By applying L'Hôpital's rule twice we get $$\lim_{x\to\infty}\frac{x^2}{e^x}=\lim_{x\to \infty}\frac{2x}{e^x}=\lim_{x\to \infty}\frac{2}{e^x}=0.$$ Another example of the usage of L'Hôpital's rule can be found here.