L'Hospital's Rule

This article is part of the MathHelp Tutoring Wiki

Definition

If ƒ and g are differentiable and $\underset{x\to c}{\lim }f(x)=\underset{x\to c}{\lim }g(x)=0$ or $\pm \mathrm{\infty }$

and $\underset{x\to c}{\lim }{f}^{\prime }(x)/g^{\prime }(x)$ exists,

then $\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\underset{x\to c}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}.$

Example

Find $\underset{t\to 0}{\lim }\frac{\exp 3t-1}{t}$

$\underset{t\to 0}{\lim }\exp 3t-1=0$

$\underset{t\to 0}{\lim }t=0$

$f(t)=\exp 3t-1$

$f^{\prime }(t)=3exp3t$

$g(t)=t$ $g^{\prime }(t)=1$

$limt->0(e^{(}3t)-1)/t=limt->0(3e^{(}3t))/1=limt->03e^{(}3t)=3$

• Back to Differential Calculus
• Back to MathHelp