L'Hopital's Rule

Theorem

1. Suppose $f$ and $g$ are differentiable on $a$
2. If $g(x)$ is non-zero at the point $a$
3. If ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ is in Indeterminate Form
4. Then, ${\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}={\lim }_{x\to a}\frac{f(x)}{g(x)}$ This law is iterative, so if you keep getting Indeterminate Form, then you can repeat this law over and over again.

Proof

• L’Hopital Rule Proof
• L’Hopital Rule Intuition

Indeterminate Form Approaches

$\frac{0}{0},\frac{\infty }{\infty }$

Simply apply l’hopital’s rule

$0\ast \infty$, $\infty \ast -\infty$

$0\ast \infty =\frac{0}{\left(\frac{1}{\infty }\right)}$

$0^0,{\infty }^0,1^{\infty }$

$L={\infty }^0$ $\ln (L)=0\ln (\infty )$ $\frac{0}{(\frac{1}{\ln (\infty )})}$ $\frac{0}{0}$