# L'Hospital rule used twice

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

• You are right to question the result. The solution you were shown is incorrect. – Umberto P. Feb 7 '18 at 18:36
• You can't use the Hospital twice on this one. – Lord Shark the Unknown Feb 7 '18 at 18:36
• Any suggestions how to compute it otherwise? – Daniel Feb 7 '18 at 20:27

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.