We have indeterminateness of type
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(5 x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(3 x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(5 x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}}{\left(3 \cot^{2}{\left(3 x \right)} + 3\right) \cot^{2}{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{3 \cot^{2}{\left(3 x \right)} + 3}}{\frac{d}{d x} \frac{\cot^{2}{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(\frac{\left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}} + \frac{\left(- 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} \cot^{2}{\left(5 x \right)} - 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} - 5 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot^{2}{\left(3 x \right)} \cot{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}\right)^{2}}\right) \left(3 \cot^{2}{\left(3 x \right)} + 3\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(\frac{\left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}} + \frac{\left(- 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} \cot^{2}{\left(5 x \right)} - 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} - 5 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot^{2}{\left(3 x \right)} \cot{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}\right)^{2}}\right) \left(3 \cot^{2}{\left(3 x \right)} + 3\right)^{2}}\right)$$
=
$$\frac{5}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)