# Thread: Find the limit of (1+x)^(3/x)

1. ## Find the limit of (1+x)^(3/x)

I'm trying to find the limit of (1+x)^(3/x) as x approaches 0. I've tried a million different ways to change the power to something where I can simplify 3/x by multiplying (1+x)^(3/x) by [(1+x)^y]/[(1+x)^y], where y is sinx or x^2/x or something else so that I can add it to 3/x and get something that I wouldn't have to divide by zero. .... I hope that doesn't sound too confusing.... Oh, and I need to know how to do this without a calculator, so if you could explain that to, that would be so totally amazing!!!

Thanks!

2. ## Re: Fidn the limit of (1+x)^(3/x)

use the fact that the limit of (1+x)^(1/x)=e.

you should get e^3.

3. ## Re: Find the limit of (1+x)^(3/x)

Originally Posted by NothingButSmiles
I'm trying to find the limit of (1+x)^(3/x) as x approaches 0. I've tried a million different ways to change the power to something where I can simplify 3/x by multiplying (1+x)^(3/x) by [(1+x)^y]/[(1+x)^y], where y is sinx or x^2/x or something else so that I can add it to 3/x and get something that I wouldn't have to divide by zero. .... I hope that doesn't sound too confusing.... Oh, and I need to know how to do this without a calculator, so if you could explain that to, that would be so totally amazing!!!

Thanks!
An alternative way is to let $t = \dfrac{1}{x} \ \Leftrightarrow \ x = \dfrac{1}{t}$. Therefore as $x \rightarrow 0^+ \text{ then } t \rightarrow \infty$

This will enable you to rewrite your limit as $\lim_{t \to \infty} \left(1+\dfrac{1}{t}\right)^{3t} = \lim_{t \to \infty} \left( \left(1+\dfrac{1}{t}\right)^{t}\right)^3$

That last limit is a classic way of defining $e$