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Expresiones semejantes
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Expresiones con funciones
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- log(|x|)
Límite de la función (-1+log(x^2))/(1+log(x^2))
Solución
Ha introducido
[src]
/ / 2\\
|-1 + log\x /|
lim |------------|
x->0+| / 2\ |
\1 + log\x / /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right)$$
Limit((-1 + log(x^2))/(1 + log(x^2)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x^{2} \right)} + 1} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x^{2} \right)} - 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)} + 1}}{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)} - 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} + 1}{\log{\left(x^{2} \right)}^{2} + 2 \log{\left(x^{2} \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)}^{2} + 2 \log{\left(x^{2} \right)} + 1}}{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} + 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}\right) \left(\frac{\log{\left(x^{2} \right)}^{4}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}^{3}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{6 \log{\left(x^{2} \right)}^{2}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{1}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}}}{\frac{d}{d x} \left(\frac{\log{\left(x^{2} \right)}^{4}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}^{3}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{6 \log{\left(x^{2} \right)}^{2}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{1}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 \log{\left(x^{2} \right)}}{x^{2}} + \frac{12}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}\right)^{2} \left(\frac{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{4}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{3}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{6 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{2}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{8 \log{\left(x^{2} \right)}^{3}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}^{2}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{8}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 \log{\left(x^{2} \right)}}{x^{2}} + \frac{12}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}\right)^{2} \left(\frac{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{4}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{3}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{6 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{2}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{8 \log{\left(x^{2} \right)}^{3}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}^{2}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{8}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)}\right)}\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
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x^{2} \right)} + 1} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x^{2} \right)} - 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)} + 1}}{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)} - 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} + 1}{\log{\left(x^{2} \right)}^{2} + 2 \log{\left(x^{2} \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)}^{2} + 2 \log{\left(x^{2} \right)} + 1}}{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} + 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}\right) \left(\frac{\log{\left(x^{2} \right)}^{4}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}^{3}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{6 \log{\left(x^{2} \right)}^{2}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{1}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}}}{\frac{d}{d x} \left(\frac{\log{\left(x^{2} \right)}^{4}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}^{3}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{6 \log{\left(x^{2} \right)}^{2}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{4 \log{\left(x^{2} \right)}}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}} + \frac{1}{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 \log{\left(x^{2} \right)}}{x^{2}} + \frac{12}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}\right)^{2} \left(\frac{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{4}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{3}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{6 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{2}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{8 \log{\left(x^{2} \right)}^{3}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}^{2}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{8}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 \log{\left(x^{2} \right)}}{x^{2}} + \frac{12}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}}{x} + \frac{4}{x}\right)^{2} \left(\frac{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{4}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{3}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{6 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}^{2}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{4 \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}\right) \log{\left(x^{2} \right)}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{- \frac{4 \log{\left(x^{2} \right)}^{5}}{x^{2}} + \frac{52 \log{\left(x^{2} \right)}^{4}}{x^{2}} - \frac{104 \log{\left(x^{2} \right)}^{3}}{x^{2}} + \frac{40 \log{\left(x^{2} \right)}^{2}}{x^{2}} + \frac{44 \log{\left(x^{2} \right)}}{x^{2}} - \frac{28}{x^{2}}}{\left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)^{2}} + \frac{8 \log{\left(x^{2} \right)}^{3}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}^{2}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{24 \log{\left(x^{2} \right)}}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)} + \frac{8}{x \left(- \frac{4 \log{\left(x^{2} \right)}^{5}}{x} + \frac{12 \log{\left(x^{2} \right)}^{4}}{x} - \frac{8 \log{\left(x^{2} \right)}^{3}}{x} - \frac{8 \log{\left(x^{2} \right)}^{2}}{x} + \frac{12 \log{\left(x^{2} \right)}}{x} - \frac{4}{x}\right)}\right)}\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]
/ / 2\\
|-1 + log\x /|
lim |------------|
x->0+| / 2\ |
\1 + log\x / /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right)$$
1
$$1$$
= 1.12115986417269
/ / 2\\
|-1 + log\x /|
lim |------------|
x->0-| / 2\ |
\1 + log\x / /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right)$$
1
$$1$$
= 1.12115986417269
= 1.12115986417269
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = 1$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = -1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = -1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = 1$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = 1$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = -1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = -1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x^{2} \right)} - 1}{\log{\left(x^{2} \right)} + 1}\right) = 1$$
Más detalles con x→-oo