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Expresiones idénticas
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- log5+x/logsin5+x
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Expresiones semejantes
- log(5+x)/log(sin(5-x))
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Expresiones con funciones
- Logaritmo log
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- Logaritmo log
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- log((1+x^2)/(1+x^2-x))/(b*x)
- log(cos(2*x))/(x*sin(sin(x)))
- log(x)/sin(x)^2
- log(-1+2*x)*log(2-2*x)
- Seno sin
- sin(x)^2*sin(2*x)^3/x
- sin(sin(x))/sin(x)
- sin(sin(7*x))^2/(1-cos(sin(5*x)))
- sin(23*x)/tan(x)
- sin((1+t^2)/t)
Límite de la función log(5+x)/log(sin(5+x))
Solución
Ha introducido
[src]
/ log(5 + x) \ lim |---------------| x->-5+\log(sin(5 + x))/
$$\lim_{x \to -5^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right)$$
Limit(log(5 + x)/log(sin(5 + x)), x, -5)
Método de l'Hopital
Tenemos la indeterminación de tipo
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)
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)
Gráfica
A la izquierda y a la derecha
[src]
/ log(5 + x) \ lim |---------------| x->-5+\log(sin(5 + x))/
$$\lim_{x \to -5^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right)$$
1
$$1$$
= 0.999999997107365
/ log(5 + x) \ lim |---------------| x->-5-\log(sin(5 + x))/
$$\lim_{x \to -5^-}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right)$$
1
$$1$$
= (0.999999999885219 - 9.46614801643366e-11j)
= (0.999999999885219 - 9.46614801643366e-11j)
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to -5^-}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = 1$$
Más detalles con x→-5 a la izquierda
$$\lim_{x \to -5^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(5 \right)}}{\log{\left(- \sin{\left(5 \right)} \right)} + i \pi}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(5 \right)}}{\log{\left(- \sin{\left(5 \right)} \right)} + i \pi}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(6 \right)}}{\log{\left(- \sin{\left(6 \right)} \right)} + i \pi}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(6 \right)}}{\log{\left(- \sin{\left(6 \right)} \right)} + i \pi}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→-oo
Más detalles con x→-5 a la izquierda
$$\lim_{x \to -5^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(5 \right)}}{\log{\left(- \sin{\left(5 \right)} \right)} + i \pi}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(5 \right)}}{\log{\left(- \sin{\left(5 \right)} \right)} + i \pi}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(6 \right)}}{\log{\left(- \sin{\left(6 \right)} \right)} + i \pi}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\log{\left(6 \right)}}{\log{\left(- \sin{\left(6 \right)} \right)} + i \pi}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 5 \right)}}{\log{\left(\sin{\left(x + 5 \right)} \right)}}\right) = \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→-oo