Solve $\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$ I am having great problems in solving this:
$$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$$
I am trying to solve this for hours, no solution in sight. I tried so many ways on my paper here, which all lead to nonsense or to nowhere. I concluded that I have to use the third binomial formula here, so my next step would be:
$$a^3-b^3=(a-b)(a^2+ab+b^2)$$ so 
$$a-b=\frac{a^3-b^3}{a^2+ab+b^2}$$
I tried expanding it as well, which led to absolutely nothing. These are my writings to this:


Consider the function $f(x)=x^{1/3}$.  By the mean value theorem there's a number $y\in (n, n+\sqrt n)$ such that
$$
f(n+\sqrt n) - f(n) = f'(y)(n+\sqrt n - n)= \frac{y^{-2/3}}{3}\sqrt n<n^{-2/3}\sqrt n=n^{-1/6}\to 0.
$$
