# Thread: help proof limit n to infinity (n+1)^1/3 - n (1/3)

1. ## help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

Please help to shed some light on how to proof this. This is a home work problem from my real analysis class.

thanks

3. ## Re: help proof limit n to infinity (n+1)^1/3 - n (1/3)

Originally Posted by MaxJasper
Try (n+1)/n = 1+1/n
How do I do that? I can't find a common factor that will turn it into the form (n+1). I have tried to multiply with [(n+1)^1/3 + n^(1/3)] / [(n+1)^1/3 + n^(1/3)] . But can't get rib of the power.

5. ## Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

Originally Posted by kanli
Please help to shed some light on how to proof this. This is a home work problem from my real analysis class.

thanks
Is this supposed to be \displaystyle \begin{align*} (n + 1)^{\frac{1}{3}} - n^{\frac{1}{3}} \end{align*}?

6. ## Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

Originally Posted by kanli
Please help to shed some light on how to proof this. This is a home work problem from my real analysis class.

thanks
Hint: Use the difference of cubes formula

$(a-b)(a^2+ab+b^2)=a^3-b^3$

This gives the limit

$\lim_{x \to \infty}\frac{(n+1)-n}{(n+1)^{\frac{2}{3}}+n^{\frac{1}{3}} (n+1)^{\frac{2}{3}}+n^{\frac{2}{3}} }$

7. ## Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

thank you very much. I solved the problem.