Do not understand L'Hopital Rule 
I have just started learning L'Hopital rule, and so far I thought I understood everything until I stumbled upon this question 
  $$\lim_{x\to 0}  \frac{\ln(\cos(ax))}{\ln(\cos(bx))}.$$

To this, eventually got
$$\lim_{x\to 0}  \frac{a \sin(ax) \cos(bx)}{b \sin(bx)\cos(ax)}$$
From my knowledge, $\sin(0)$ is 0!! and the whole thing will be ''$\frac{0}{0}$'', however the answer key I was given does not continue the implementation of the L'Hopital rule, but instead obtains the answer $\frac{a^2}{b^2}$.
Is there some important concept I'm missing out? Or is the differentiation supposed to continue and the answer key just skipped the steps?

$$\lim_{x\to}\dfrac{\sin ax}{\sin bx}\cdot\dfrac{\cos bx}{\cos ax}$$
$$=\dfrac ab\lim_{x\to0}\dfrac{\sin ax}{ax}\cdot\lim_{x\to0}\dfrac{bx}{\sin bx}\lim_{x\to0}\dfrac{\cos bx}{\cos ax}=?$$
Mind you $\lim_{x\to0}\implies x\ne0$
