L'Hospital's Rule

Definition

Suppose $f,g:\mathcal{D}\to \mathbb{R}$ is differentiable on the $\mathcal{D}-\{a\}$, $\underset{x\to a}{\lim }f(x)=\underset{x\to a}{\lim }g(x)=0$ or $\underset{x\to a}{\lim }|f(x)|={\lim }_{x\to a}|g(x)|=\infty$, $\forall x\in \mathcal{D}-\{a\},g^{\prime }(x)\ne 0$, and $\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$ exists, then $\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}=L\Rightarrow \underset{x\to a}{\lim }\frac{f(x)}{g(x)}=L$ where $L\in \mathbb{R}\cup \{-\infty ,\infty \}$