L'Hopital's Rule

#calc/theorem

• A technique used to evaluate limits that take indeterminate forms: $0/0$ or $\infty /\infty$.

Statement

The rule states that for for functions $f$ and $g$ where ${\displaystyle \underset{x\to a}{lim}\frac{f(x)}{g(x)}=\frac{0}{0}}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$, ${\displaystyle \underset{x\to a}{lim}\frac{f(x)}{g(x)}=\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}}$.

• Essentially, if it’s an indeterminate limit, derive the top and bottom.
• Make sure $f(x)$ and $g(x)$ are differentiable and the new limit exists

Examples

1. ${\displaystyle \underset{x\to 0}{lim}\frac{\sin x}{x}=\underset{x\to 0}{lim}\frac{\cos x}{1}=1}$
2. ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x}{e^x}=\underset{x\to \mathrm{\infty }}{lim}\frac{1}{e^x}=0}$