Thread: 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!

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

you should get e^3.

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 . Therefore as


This will enable you to rewrite your limit as


That last limit is a classic way of defining