Find the value of the following limit.
![\[ \lim_{x \to 0} \frac{\log(\cos (ax))}{\log (\cos (bx))}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1a328400a70e2996f2a2345b19a92b4e_l3.png "Rendered by QuickLaTeX.com")
Applying L’Hopital’s rule we have,
![\begin{align*} \lim_{x \to 0} \frac{\log(\cos (ax))}{\log(\cos (bx))} &= \lim_{x \to 0} \left( \frac{a}{b} \cdot \frac{\cos (bx)}{\cos (ax)} \cdot \frac{\sin (ax)}{\sin (bx)} \right) \\[9pt] &= \frac{a}{b} \lim_{x \to 0} \frac{a \cos (ax)}{b \cos (bx)} \\[9pt] &= \left(\frac{a}{b} \right)^2. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-fad738378072276b7e4f2070ca046c04_l3.png "Rendered by QuickLaTeX.com")
(We used that 
from this exercise, Section 7.11, Exercise #1). Since this limit exists, the application of L’Hopital’s rule was justified.
Find the value of the following limit.
Applying L’Hopital’s rule we have,
(We used that from this exercise, Section 7.11, Exercise #1). Since this limit exists, the application of L’Hopital’s rule was justified.