Tenemos la indeterminación de tipo
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \operatorname{atan}{\left(4 x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(5 x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\cot{\left(5 x \right)} \operatorname{atan}{\left(4 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{atan}{\left(4 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(5 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 \cot^{2}{\left(5 x \right)}}{\left(16 x^{2} + 1\right) \left(5 \cot^{2}{\left(5 x \right)} + 5\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 \cot^{2}{\left(5 x \right)}}{5 \cot^{2}{\left(5 x \right)} + 5}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{5 \cot^{2}{\left(5 x \right)} + 5}}{\frac{d}{d x} \frac{1}{4 \cot^{2}{\left(5 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(10 \cot^{2}{\left(5 x \right)} + 10\right) \left(- \frac{25 \cot^{4}{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}} - \frac{50 \cot^{2}{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}} - \frac{25}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{10 \cot^{2}{\left(5 x \right)} + 10}}{\frac{d}{d x} \left(- \frac{25 \cot^{4}{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}} - \frac{50 \cot^{2}{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}} - \frac{25}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{10 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{\left(10 \cot^{2}{\left(5 x \right)} + 10\right)^{2} \left(- \frac{25 \left(200 \left(- 30 \cot^{2}{\left(5 x \right)} - 30\right) \cot^{5}{\left(5 x \right)} + 200 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}\right) \cot^{4}{\left(5 x \right)}}{\left(- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}\right)^{2}} - \frac{50 \left(200 \left(- 30 \cot^{2}{\left(5 x \right)} - 30\right) \cot^{5}{\left(5 x \right)} + 200 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}\right)^{2}} - \frac{25 \left(200 \left(- 30 \cot^{2}{\left(5 x \right)} - 30\right) \cot^{5}{\left(5 x \right)} + 200 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}\right)}{\left(- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}\right)^{2}} - \frac{25 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}} - \frac{50 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{10 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{\left(10 \cot^{2}{\left(5 x \right)} + 10\right)^{2} \left(- \frac{25 \left(200 \left(- 30 \cot^{2}{\left(5 x \right)} - 30\right) \cot^{5}{\left(5 x \right)} + 200 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}\right) \cot^{4}{\left(5 x \right)}}{\left(- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}\right)^{2}} - \frac{50 \left(200 \left(- 30 \cot^{2}{\left(5 x \right)} - 30\right) \cot^{5}{\left(5 x \right)} + 200 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}\right)^{2}} - \frac{25 \left(200 \left(- 30 \cot^{2}{\left(5 x \right)} - 30\right) \cot^{5}{\left(5 x \right)} + 200 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}\right)}{\left(- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}\right)^{2}} - \frac{25 \left(- 20 \cot^{2}{\left(5 x \right)} - 20\right) \cot^{3}{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}} - \frac{50 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{- 200 \cot^{6}{\left(5 x \right)} - 200 \cot^{4}{\left(5 x \right)}}\right)}\right)$$
=
$$\frac{4}{5}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)