It's easy to find the limit by L'Hopital's rule, but that's the second part of the question after proving the limit exists.Verify the following limit exists:

$\displaystyle \lim_{x \rightarrow 0} \frac{(1+\frac{x}{2})log(1+x)-x}{x^3}$

Therefore, is there a way to prove that the limit exists without using L'hopitals rule (or is it acceptable to just find the limit by L'Hopitals rule and say that it must exist since we found it!)?