Answer
$$\frac{4}{5}$$
Work Step by Step
We have
$$
\lim _{x \rightarrow 0} \frac{\tan 4x}{\tan 5x} =\frac{0}{0}
$$
is an intermediate form, then we can apply L’Hôpital’s Rule as follows
$$
\lim _{x \rightarrow 0} \frac{\tan 4x}{\tan 5x} =
\lim _{x \rightarrow 0} \frac{4\sec^2 4x}{5\sec^2 5x}= \frac{4}{5}.
$$
