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.

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- Dec 2nd 2006, 09:17 AMnanotek887L'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. - Dec 2nd 2006, 09:30 AMTD!
You need to get an indeterminate form such as 0/0 or inf/inf to apply l'Hôpital.

$\displaystyle \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:

$\displaystyle

\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:

$\displaystyle

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

$