Tenemos la indeterminación de tipo
0/0,
tal que el límite para el numerador es
$$\lim_{x \to -5^+} \frac{1}{\log{\left(\sin{\left(x + 5 \right)} \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to -5^+} \frac{1}{\log{\left(x + 5 \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to -5^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right)$$
=
$$\lim_{x \to -5^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x + 5 \right)} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(x + 5 \right)}}}\right)$$
=
$$\lim_{x \to -5^+}\left(\frac{\left(x + 5\right) \log{\left(x + 5 \right)}^{2} \cos{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}^{2} \sin{\left(x + 5 \right)}}\right)$$
=
$$\lim_{x \to -5^+}\left(\frac{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}}{\log{\left(\sin{\left(x + 5 \right)} \right)}^{2} \sin{\left(x + 5 \right)}}\right)$$
=
$$\lim_{x \to -5^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x + 5 \right)} \right)}^{2}}}{\frac{d}{d x} \frac{\sin{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}}}\right)$$
=
$$\lim_{x \to -5^+}\left(- \frac{2 \cos{\left(x + 5 \right)}}{\left(\frac{\cos{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}} + \frac{\left(- \frac{2 x \log{\left(x + 5 \right)}}{x + 5} - \log{\left(x + 5 \right)}^{2} - \frac{10 \log{\left(x + 5 \right)}}{x + 5}\right) \sin{\left(x + 5 \right)}}{\left(x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}\right)^{2}}\right) \log{\left(\sin{\left(x + 5 \right)} \right)}^{3} \sin{\left(x + 5 \right)}}\right)$$
=
$$\lim_{x \to -5^+}\left(- \frac{2}{\left(- \frac{2 x \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{10 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{\log{\left(x + 5 \right)}^{2} \sin{\left(x + 5 \right)}}{x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}} + \frac{\cos{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}}\right) \log{\left(\sin{\left(x + 5 \right)} \right)}^{3} \sin{\left(x + 5 \right)}}\right)$$
=
$$\lim_{x \to -5^+}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2 x \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{10 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{\log{\left(x + 5 \right)}^{2} \sin{\left(x + 5 \right)}}{x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}} + \frac{\cos{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}}}}{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(x + 5 \right)} \right)}^{3} \sin{\left(x + 5 \right)}}{2}\right)}\right)$$
=
$$\lim_{x \to -5^+}\left(\frac{\frac{2 x \log{\left(x + 5 \right)} \cos{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} + \frac{2 x \left(- \frac{4 x^{3} \log{\left(x + 5 \right)}^{3}}{x + 5} - 3 x^{2} \log{\left(x + 5 \right)}^{4} - \frac{60 x^{2} \log{\left(x + 5 \right)}^{3}}{x + 5} - 30 x \log{\left(x + 5 \right)}^{4} - \frac{300 x \log{\left(x + 5 \right)}^{3}}{x + 5} - 75 \log{\left(x + 5 \right)}^{4} - \frac{500 \log{\left(x + 5 \right)}^{3}}{x + 5}\right) \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{\left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)^{2}} + \frac{2 x \sin{\left(x + 5 \right)}}{\left(x + 5\right) \left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)} + \frac{2 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} + \frac{10 \log{\left(x + 5 \right)} \cos{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} + \frac{10 \left(- \frac{4 x^{3} \log{\left(x + 5 \right)}^{3}}{x + 5} - 3 x^{2} \log{\left(x + 5 \right)}^{4} - \frac{60 x^{2} \log{\left(x + 5 \right)}^{3}}{x + 5} - 30 x \log{\left(x + 5 \right)}^{4} - \frac{300 x \log{\left(x + 5 \right)}^{3}}{x + 5} - 75 \log{\left(x + 5 \right)}^{4} - \frac{500 \log{\left(x + 5 \right)}^{3}}{x + 5}\right) \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{\left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)^{2}} + \frac{\log{\left(x + 5 \right)}^{2} \cos{\left(x + 5 \right)}}{x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}} + \frac{\left(- \frac{4 x^{2} \log{\left(x + 5 \right)}^{3}}{x + 5} - 2 x \log{\left(x + 5 \right)}^{4} - \frac{40 x \log{\left(x + 5 \right)}^{3}}{x + 5} - 10 \log{\left(x + 5 \right)}^{4} - \frac{100 \log{\left(x + 5 \right)}^{3}}{x + 5}\right) \log{\left(x + 5 \right)}^{2} \sin{\left(x + 5 \right)}}{\left(x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}\right)^{2}} + \frac{\sin{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}} - \frac{\left(- \frac{2 x \log{\left(x + 5 \right)}}{x + 5} - \log{\left(x + 5 \right)}^{2} - \frac{10 \log{\left(x + 5 \right)}}{x + 5}\right) \cos{\left(x + 5 \right)}}{\left(x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}\right)^{2}} + \frac{10 \sin{\left(x + 5 \right)}}{\left(x + 5\right) \left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)} + \frac{2 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{\left(x + 5\right) \left(x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}\right)}}{\left(- \frac{\log{\left(\sin{\left(x + 5 \right)} \right)}^{3} \cos{\left(x + 5 \right)}}{2} - \frac{3 \log{\left(\sin{\left(x + 5 \right)} \right)}^{2} \cos{\left(x + 5 \right)}}{2}\right) \left(- \frac{2 x \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{10 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{\log{\left(x + 5 \right)}^{2} \sin{\left(x + 5 \right)}}{x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}} + \frac{\cos{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}}\right)^{2}}\right)$$
=
$$\lim_{x \to -5^+}\left(\frac{\frac{2 x \log{\left(x + 5 \right)} \cos{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} + \frac{2 x \left(- \frac{4 x^{3} \log{\left(x + 5 \right)}^{3}}{x + 5} - 3 x^{2} \log{\left(x + 5 \right)}^{4} - \frac{60 x^{2} \log{\left(x + 5 \right)}^{3}}{x + 5} - 30 x \log{\left(x + 5 \right)}^{4} - \frac{300 x \log{\left(x + 5 \right)}^{3}}{x + 5} - 75 \log{\left(x + 5 \right)}^{4} - \frac{500 \log{\left(x + 5 \right)}^{3}}{x + 5}\right) \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{\left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)^{2}} + \frac{2 x \sin{\left(x + 5 \right)}}{\left(x + 5\right) \left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)} + \frac{2 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} + \frac{10 \log{\left(x + 5 \right)} \cos{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} + \frac{10 \left(- \frac{4 x^{3} \log{\left(x + 5 \right)}^{3}}{x + 5} - 3 x^{2} \log{\left(x + 5 \right)}^{4} - \frac{60 x^{2} \log{\left(x + 5 \right)}^{3}}{x + 5} - 30 x \log{\left(x + 5 \right)}^{4} - \frac{300 x \log{\left(x + 5 \right)}^{3}}{x + 5} - 75 \log{\left(x + 5 \right)}^{4} - \frac{500 \log{\left(x + 5 \right)}^{3}}{x + 5}\right) \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{\left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)^{2}} + \frac{\log{\left(x + 5 \right)}^{2} \cos{\left(x + 5 \right)}}{x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}} + \frac{\left(- \frac{4 x^{2} \log{\left(x + 5 \right)}^{3}}{x + 5} - 2 x \log{\left(x + 5 \right)}^{4} - \frac{40 x \log{\left(x + 5 \right)}^{3}}{x + 5} - 10 \log{\left(x + 5 \right)}^{4} - \frac{100 \log{\left(x + 5 \right)}^{3}}{x + 5}\right) \log{\left(x + 5 \right)}^{2} \sin{\left(x + 5 \right)}}{\left(x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}\right)^{2}} + \frac{\sin{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}} - \frac{\left(- \frac{2 x \log{\left(x + 5 \right)}}{x + 5} - \log{\left(x + 5 \right)}^{2} - \frac{10 \log{\left(x + 5 \right)}}{x + 5}\right) \cos{\left(x + 5 \right)}}{\left(x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}\right)^{2}} + \frac{10 \sin{\left(x + 5 \right)}}{\left(x + 5\right) \left(x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}\right)} + \frac{2 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{\left(x + 5\right) \left(x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}\right)}}{\left(- \frac{\log{\left(\sin{\left(x + 5 \right)} \right)}^{3} \cos{\left(x + 5 \right)}}{2} - \frac{3 \log{\left(\sin{\left(x + 5 \right)} \right)}^{2} \cos{\left(x + 5 \right)}}{2}\right) \left(- \frac{2 x \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{10 \log{\left(x + 5 \right)} \sin{\left(x + 5 \right)}}{x^{3} \log{\left(x + 5 \right)}^{4} + 15 x^{2} \log{\left(x + 5 \right)}^{4} + 75 x \log{\left(x + 5 \right)}^{4} + 125 \log{\left(x + 5 \right)}^{4}} - \frac{\log{\left(x + 5 \right)}^{2} \sin{\left(x + 5 \right)}}{x^{2} \log{\left(x + 5 \right)}^{4} + 10 x \log{\left(x + 5 \right)}^{4} + 25 \log{\left(x + 5 \right)}^{4}} + \frac{\cos{\left(x + 5 \right)}}{x \log{\left(x + 5 \right)}^{2} + 5 \log{\left(x + 5 \right)}^{2}}\right)^{2}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)