L'Hospital rule used twice

Question

I need to compute the limit of this expression: $$\lim_{x\to 0} \frac{a^x - b^x}{cx^3 + dx^2}$$. In the solution (given, in the link) they used L'Hospital twice. I understand the first time, but in the second time I can't see why (and how could they) use it. i.e the conditions weren't there... am I missing something?

solution to the problem

Answer 1

Good job! You are absolutely correct to question this solution.

In the second limit, the limit of the denominator is $$\lim_{x \rightarrow 0} \left(3cx^2 + 2dx\right) = 0,$$ while the limit of the numerator is $$\lim_{x \rightarrow 0} \left(\ln(b) b^x - \ln(a) a^x\right) = \ln(b) - \ln(a),$$ which is not zero unless $a = b$. This means that the second application of l'Hôpital's rule in the solution is invalid.

Even seasoned instructors sometimes make mistakes when preparing examples or solutions. It's good to verify things for yourself, as you have wisely done in this case.