Math Help - limit question

1. limit question

lim_{x→0+} (1+3x)^(1/x) = e³

How do I fill in the blanks? In other words, how do I show that they are equal?

Thank you!

2. Re: limit question

If $L=\lim_{x\to a}f(x)^{g(x)}$ presents an indetermination of the form $1^{\infty}$ then, by a well known result $L=e^{\lambda}$ being $\lambda=\lim_{x\to a}(f(x)-1)g(x)$ .

3. Re: limit question

Thank you! I'm pretty sure I recognize those results by using them in differential equations.

4. Re: limit question

Originally Posted by CountingPenguins
lim_{x→0+} (1+3x)^(1/x) = e³

How do I fill in the blanks? In other words, how do I show that they are equal?

Thank you!
A well known definition of e is $\lim_{t \to +\infty} \left(1 + \frac{1}{t}\right)^t$.

So make the substitution $3x = \frac{1}{t}$ in your limit ....