How to evaluate the limit $\lim_{x\to 0} ((1+2x)^{1/3 }-1)/ x$ without using the l'Hospital's rule? $$\lim_{x\to 0} \frac{(1+2x)^{\frac{1}{3}}-1}{x}.$$
Please do not use the l'hospital's rule as I am trying to solve this limit without using that rule... to no avail...

Hints:
As $x\to 0$, $\sqrt[3]{1+2x}-1\sim \frac23x$

In fact: $\sqrt[n]{1+x}-1\sim \frac xn$ as $x\to 0$.

