Limit of multiple third roots without L'Hospital I'm not able to solve the limit of a textbook question.
The limit:
$$\lim_{x\to \infty} (\sqrt[3]{x^2}(\sqrt[3]{x+1} - \sqrt[3]{x}))$$
I've been able to simplify the limit to:
$$\lim_{x\to \infty} (\sqrt[3]{x^3+x^2} - x)$$
How do I solve this limit?
Note: no L'Hospital allowed.

Using
$$a-b=\frac{a^3-b^3}{a^2+ab+b^2}$$
gives
$$\sqrt[3]{x^3+x^2}-x=\frac{x^2}{(x^3+x^2)^{2/3}+x(x^3+x^2)^{1/3}+x^2}
=\frac{1}{(1+x^{-1})^{2/3}+(1+x^{-1})^{1/3}+1}\to\frac13$$
as $x\to\infty$.
