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

Question

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

Answer 1

Hint:

If $a \ne b$ then use L'Hopital:

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

It is necessary analyse the relation between $a$ and $b$ ($a>b$ or $a

Answer 2

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

Answer 3

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.