L'Hospital's Rule

Definition

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

and ${\displaystyle \lim _{x\to c}f'(x)/g'(x)}$ exists,

then ${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}$

Example

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

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

${\displaystyle \lim _{t\to 0}t=0}$

${\displaystyle f(t)=\exp {3t}-1}$

${\displaystyle f'(t)=3exp{3t}}$

${\displaystyle g(t)=t}$ ${\displaystyle g'(t)=1}$

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