$\lim_{x \to 0} \frac{a^x - b^x}{cx^3 + dx^2}$ 
$$\lim_{x \to 0} \frac{a^x - b^x}{cx^3 + dx^2}$$
  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

You are correct left sided limit is -infinity & right sided limit is  infinity. You can also apply L hospital rule to check
