# Math Help - L'Hopital's Rule

1. ## L'Hopital's Rule

Could someone help me use L'Hopital's Rule to find the following limit?

lim (as x approaches 0) (1+(3/x))^x
Thank you.

2. You need to get an indeterminate form such as 0/0 or inf/inf to apply l'Hôpital.

$\left( {1 + \frac{3}{x}} \right)^x = e^{\ln \left( {1 + \frac{3}{x}} \right)^x } = e^{x\ln \left( {1 + \frac{3}{x}} \right)}$

Now we can use the fact that:

$
\mathop {\lim }\limits_{x \to 0} e^{x\ln \left( {1 + \frac{3}{x}} \right)} = e^{\mathop {\lim }\limits_{x \to 0} x\ln \left( {1 + \frac{3}{x}} \right)}
$

Now you have two possibilities, one is:

$
x\ln \left( {1 + \frac{3}{x}} \right) = \frac{{\ln \left( {1 + \frac{3}{x}} \right)}}{{\frac{1}{x}}}
$