$\lim_{x \to 0} \frac{a^x - b^x}{cx^3 + dx^2}$ 
$$\lim_{x \to 0} \frac{a^x - b^x}{cx^3 + dx^2}$$
  where $a,b>0$ and $c^2 + d^2 >0$.

I think that the limit does not exist as the one sided limits of 0 goes to plus inifnity and minus inifinity, but I'm not sure.
Any help appriciated

It is $0/0$ indeterminate so we may apply L'Hospital's rule to get
$$\lim_{x\to0}\frac{\ln(a)a^x-\ln(b)b^x}{3cx^2+2dx}$$
Which is now of the form $p/0$, so it diverges to $\pm\infty$.
By seeing it must be positive, we conclude it diverges to $+\infty$ if $a>b$ and negative infinity in the other case.
