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