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Límite de (-1+x^3)/(-1+x)
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Límite de x^sin(x)
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Expresiones idénticas
- uno /log(x)- uno /(- uno +x)
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- uno dividir por logaritmo de (x) menos uno dividir por ( menos uno más x)
- 1/logx-1/-1+x
- 1 dividir por log(x)-1 dividir por (-1+x)
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Expresiones semejantes
- 1/log(x)+1/(-1+x)
- 1/log(x)-1/(-1-x)
- 1/log(x)-1/(1+x)
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Expresiones con funciones
- Logaritmo log
- log(x)/log(sin(x))
- log(-1+x)/cot(pi*x)
- log(1+x)/asin(x)
- log(x)/(-2+x+x^2)
- log(x/cot(x))
Límite de la función 1/log(x)-1/(-1+x)
Solución
Ha introducido
[src]
/ 1 1 \ lim |------ - ------| x->0+\log(x) -1 + x/
$$\lim_{x \to 0^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right)$$
Limit(1/log(x) - 1/(-1 + x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(\frac{1}{\left(x - 1\right) \log{\left(x \right)}}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{x - \log{\left(x \right)} - 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{x - \log{\left(x \right)} - 1}{\left(x - 1\right) \log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\left(x - 1\right) \log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{x - \log{\left(x \right)} - 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- \frac{1}{\left(x - 1\right)^{2} \log{\left(x \right)}} - \frac{1}{x \left(x - 1\right) \log{\left(x \right)}^{2}}\right) \left(x - \log{\left(x \right)} - 1\right)^{2}}{-1 + \frac{1}{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\left(x - \log{\left(x \right)} - 1\right)^{2}}{-1 + \frac{1}{x}}}{\frac{d}{d x} \frac{1}{- \frac{1}{\left(x - 1\right)^{2} \log{\left(x \right)}} - \frac{1}{x \left(x - 1\right) \log{\left(x \right)}^{2}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2 x^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{2 x}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} - \frac{2 x}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4 x \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{4 x}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2 x}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} + \frac{4 x}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} + \frac{1}{x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x + 1} + \frac{2 \log{\left(x \right)}^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{2} + 5 x^{4} \log{\left(x \right)}^{2} - 10 x^{3} \log{\left(x \right)}^{2} + 10 x^{2} \log{\left(x \right)}^{2} - 5 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{- x^{5} \log{\left(x \right)} + 5 x^{4} \log{\left(x \right)} - 10 x^{3} \log{\left(x \right)} + 10 x^{2} \log{\left(x \right)} - 5 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} - \frac{2}{- x^{4} \log{\left(x \right)} + 5 x^{3} \log{\left(x \right)} - 10 x^{2} \log{\left(x \right)} + 10 x \log{\left(x \right)} - 5 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{x}} + \frac{1}{x^{6} \log{\left(x \right)}^{4} - 4 x^{5} \log{\left(x \right)}^{4} + 6 x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 4 x^{5} \log{\left(x \right)}^{3} + 6 x^{4} \log{\left(x \right)}^{3} - 4 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 4 x^{5} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(x \right)}^{2} - 4 x^{3} \log{\left(x \right)}^{2} + x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{5} \log{\left(x \right)}^{4} - 4 x^{4} \log{\left(x \right)}^{4} + 6 x^{3} \log{\left(x \right)}^{4} - 4 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{x^{5} \log{\left(x \right)}^{3} - 4 x^{4} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(x \right)}^{3} - 4 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{1}{x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + 6 x^{2} \log{\left(x \right)}^{4} - 4 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}} - \frac{4 \log{\left(x \right)}}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} - \frac{8}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{4} + 3 x^{4} \log{\left(x \right)}^{4} - 3 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{- x^{5} \log{\left(x \right)}^{3} + 3 x^{4} \log{\left(x \right)}^{3} - 3 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} - \frac{4}{- x^{4} \log{\left(x \right)}^{4} + 3 x^{3} \log{\left(x \right)}^{4} - 3 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{- x^{4} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(x \right)}^{3} - 3 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{2}{- x^{3} \log{\left(x \right)}^{4} + 3 x^{2} \log{\left(x \right)}^{4} - 3 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}}}{- \frac{2}{x^{3} \log{\left(x \right)} - 3 x^{2} \log{\left(x \right)} + 3 x \log{\left(x \right)} - \log{\left(x \right)}} - \frac{2}{x^{3} \log{\left(x \right)}^{2} - 2 x^{2} \log{\left(x \right)}^{2} + x \log{\left(x \right)}^{2}} - \frac{2}{x^{3} \log{\left(x \right)}^{3} - x^{2} \log{\left(x \right)}^{3}} - \frac{1}{x^{3} \log{\left(x \right)}^{2} - x^{2} \log{\left(x \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2 x^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{2 x}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} - \frac{2 x}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4 x \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{4 x}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2 x}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} + \frac{4 x}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} + \frac{1}{x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x + 1} + \frac{2 \log{\left(x \right)}^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{2} + 5 x^{4} \log{\left(x \right)}^{2} - 10 x^{3} \log{\left(x \right)}^{2} + 10 x^{2} \log{\left(x \right)}^{2} - 5 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{- x^{5} \log{\left(x \right)} + 5 x^{4} \log{\left(x \right)} - 10 x^{3} \log{\left(x \right)} + 10 x^{2} \log{\left(x \right)} - 5 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} - \frac{2}{- x^{4} \log{\left(x \right)} + 5 x^{3} \log{\left(x \right)} - 10 x^{2} \log{\left(x \right)} + 10 x \log{\left(x \right)} - 5 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{x}} + \frac{1}{x^{6} \log{\left(x \right)}^{4} - 4 x^{5} \log{\left(x \right)}^{4} + 6 x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 4 x^{5} \log{\left(x \right)}^{3} + 6 x^{4} \log{\left(x \right)}^{3} - 4 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 4 x^{5} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(x \right)}^{2} - 4 x^{3} \log{\left(x \right)}^{2} + x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{5} \log{\left(x \right)}^{4} - 4 x^{4} \log{\left(x \right)}^{4} + 6 x^{3} \log{\left(x \right)}^{4} - 4 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{x^{5} \log{\left(x \right)}^{3} - 4 x^{4} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(x \right)}^{3} - 4 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{1}{x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + 6 x^{2} \log{\left(x \right)}^{4} - 4 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}} - \frac{4 \log{\left(x \right)}}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} - \frac{8}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{4} + 3 x^{4} \log{\left(x \right)}^{4} - 3 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{- x^{5} \log{\left(x \right)}^{3} + 3 x^{4} \log{\left(x \right)}^{3} - 3 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} - \frac{4}{- x^{4} \log{\left(x \right)}^{4} + 3 x^{3} \log{\left(x \right)}^{4} - 3 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{- x^{4} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(x \right)}^{3} - 3 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{2}{- x^{3} \log{\left(x \right)}^{4} + 3 x^{2} \log{\left(x \right)}^{4} - 3 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}}}{- \frac{2}{x^{3} \log{\left(x \right)} - 3 x^{2} \log{\left(x \right)} + 3 x \log{\left(x \right)} - \log{\left(x \right)}} - \frac{2}{x^{3} \log{\left(x \right)}^{2} - 2 x^{2} \log{\left(x \right)}^{2} + x \log{\left(x \right)}^{2}} - \frac{2}{x^{3} \log{\left(x \right)}^{3} - x^{2} \log{\left(x \right)}^{3}} - \frac{1}{x^{3} \log{\left(x \right)}^{2} - x^{2} \log{\left(x \right)}^{2}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(\frac{1}{\left(x - 1\right) \log{\left(x \right)}}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{x - \log{\left(x \right)} - 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{x - \log{\left(x \right)} - 1}{\left(x - 1\right) \log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\left(x - 1\right) \log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{x - \log{\left(x \right)} - 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- \frac{1}{\left(x - 1\right)^{2} \log{\left(x \right)}} - \frac{1}{x \left(x - 1\right) \log{\left(x \right)}^{2}}\right) \left(x - \log{\left(x \right)} - 1\right)^{2}}{-1 + \frac{1}{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\left(x - \log{\left(x \right)} - 1\right)^{2}}{-1 + \frac{1}{x}}}{\frac{d}{d x} \frac{1}{- \frac{1}{\left(x - 1\right)^{2} \log{\left(x \right)}} - \frac{1}{x \left(x - 1\right) \log{\left(x \right)}^{2}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2 x^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{2 x}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} - \frac{2 x}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4 x \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{4 x}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2 x}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} + \frac{4 x}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} + \frac{1}{x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x + 1} + \frac{2 \log{\left(x \right)}^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{2} + 5 x^{4} \log{\left(x \right)}^{2} - 10 x^{3} \log{\left(x \right)}^{2} + 10 x^{2} \log{\left(x \right)}^{2} - 5 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{- x^{5} \log{\left(x \right)} + 5 x^{4} \log{\left(x \right)} - 10 x^{3} \log{\left(x \right)} + 10 x^{2} \log{\left(x \right)} - 5 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} - \frac{2}{- x^{4} \log{\left(x \right)} + 5 x^{3} \log{\left(x \right)} - 10 x^{2} \log{\left(x \right)} + 10 x \log{\left(x \right)} - 5 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{x}} + \frac{1}{x^{6} \log{\left(x \right)}^{4} - 4 x^{5} \log{\left(x \right)}^{4} + 6 x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 4 x^{5} \log{\left(x \right)}^{3} + 6 x^{4} \log{\left(x \right)}^{3} - 4 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 4 x^{5} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(x \right)}^{2} - 4 x^{3} \log{\left(x \right)}^{2} + x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{5} \log{\left(x \right)}^{4} - 4 x^{4} \log{\left(x \right)}^{4} + 6 x^{3} \log{\left(x \right)}^{4} - 4 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{x^{5} \log{\left(x \right)}^{3} - 4 x^{4} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(x \right)}^{3} - 4 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{1}{x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + 6 x^{2} \log{\left(x \right)}^{4} - 4 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}} - \frac{4 \log{\left(x \right)}}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} - \frac{8}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{4} + 3 x^{4} \log{\left(x \right)}^{4} - 3 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{- x^{5} \log{\left(x \right)}^{3} + 3 x^{4} \log{\left(x \right)}^{3} - 3 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} - \frac{4}{- x^{4} \log{\left(x \right)}^{4} + 3 x^{3} \log{\left(x \right)}^{4} - 3 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{- x^{4} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(x \right)}^{3} - 3 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{2}{- x^{3} \log{\left(x \right)}^{4} + 3 x^{2} \log{\left(x \right)}^{4} - 3 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}}}{- \frac{2}{x^{3} \log{\left(x \right)} - 3 x^{2} \log{\left(x \right)} + 3 x \log{\left(x \right)} - \log{\left(x \right)}} - \frac{2}{x^{3} \log{\left(x \right)}^{2} - 2 x^{2} \log{\left(x \right)}^{2} + x \log{\left(x \right)}^{2}} - \frac{2}{x^{3} \log{\left(x \right)}^{3} - x^{2} \log{\left(x \right)}^{3}} - \frac{1}{x^{3} \log{\left(x \right)}^{2} - x^{2} \log{\left(x \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2 x^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{2 x}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} - \frac{2 x}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4 x \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} - \frac{4 x}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2 x}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} + \frac{4 x}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 6 x^{5} \log{\left(x \right)}^{2} + 15 x^{4} \log{\left(x \right)}^{2} - 20 x^{3} \log{\left(x \right)}^{2} + 15 x^{2} \log{\left(x \right)}^{2} - 6 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{x^{6} \log{\left(x \right)} - 6 x^{5} \log{\left(x \right)} + 15 x^{4} \log{\left(x \right)} - 20 x^{3} \log{\left(x \right)} + 15 x^{2} \log{\left(x \right)} - 6 x \log{\left(x \right)} + \log{\left(x \right)}} + \frac{1}{x^{6} - 6 x^{5} + 15 x^{4} - 20 x^{3} + 15 x^{2} - 6 x + 1} + \frac{2 \log{\left(x \right)}^{2}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 5 x^{5} \log{\left(x \right)}^{3} + 10 x^{4} \log{\left(x \right)}^{3} - 10 x^{3} \log{\left(x \right)}^{3} + 5 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{2} + 5 x^{4} \log{\left(x \right)}^{2} - 10 x^{3} \log{\left(x \right)}^{2} + 10 x^{2} \log{\left(x \right)}^{2} - 5 x \log{\left(x \right)}^{2} + \log{\left(x \right)}^{2}} + \frac{2}{- x^{5} \log{\left(x \right)} + 5 x^{4} \log{\left(x \right)} - 10 x^{3} \log{\left(x \right)} + 10 x^{2} \log{\left(x \right)} - 5 x \log{\left(x \right)} + \log{\left(x \right)}} - \frac{4}{- x^{4} \log{\left(x \right)}^{2} + 5 x^{3} \log{\left(x \right)}^{2} - 10 x^{2} \log{\left(x \right)}^{2} + 10 x \log{\left(x \right)}^{2} - 5 \log{\left(x \right)}^{2} + \frac{\log{\left(x \right)}^{2}}{x}} - \frac{2}{- x^{4} \log{\left(x \right)} + 5 x^{3} \log{\left(x \right)} - 10 x^{2} \log{\left(x \right)} + 10 x \log{\left(x \right)} - 5 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{x}} + \frac{1}{x^{6} \log{\left(x \right)}^{4} - 4 x^{5} \log{\left(x \right)}^{4} + 6 x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{x^{6} \log{\left(x \right)}^{3} - 4 x^{5} \log{\left(x \right)}^{3} + 6 x^{4} \log{\left(x \right)}^{3} - 4 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} + \frac{1}{x^{6} \log{\left(x \right)}^{2} - 4 x^{5} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(x \right)}^{2} - 4 x^{3} \log{\left(x \right)}^{2} + x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{5} \log{\left(x \right)}^{4} - 4 x^{4} \log{\left(x \right)}^{4} + 6 x^{3} \log{\left(x \right)}^{4} - 4 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{x^{5} \log{\left(x \right)}^{3} - 4 x^{4} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(x \right)}^{3} - 4 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{4 \log{\left(x \right)}}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{4}{- x^{5} \log{\left(x \right)}^{3} + 4 x^{4} \log{\left(x \right)}^{3} - 6 x^{3} \log{\left(x \right)}^{3} + 4 x^{2} \log{\left(x \right)}^{3} - x \log{\left(x \right)}^{3}} + \frac{1}{x^{4} \log{\left(x \right)}^{4} - 4 x^{3} \log{\left(x \right)}^{4} + 6 x^{2} \log{\left(x \right)}^{4} - 4 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}} - \frac{4 \log{\left(x \right)}}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} - \frac{8}{- x^{4} \log{\left(x \right)}^{3} + 4 x^{3} \log{\left(x \right)}^{3} - 6 x^{2} \log{\left(x \right)}^{3} + 4 x \log{\left(x \right)}^{3} - \log{\left(x \right)}^{3}} + \frac{2}{- x^{5} \log{\left(x \right)}^{4} + 3 x^{4} \log{\left(x \right)}^{4} - 3 x^{3} \log{\left(x \right)}^{4} + x^{2} \log{\left(x \right)}^{4}} + \frac{2}{- x^{5} \log{\left(x \right)}^{3} + 3 x^{4} \log{\left(x \right)}^{3} - 3 x^{3} \log{\left(x \right)}^{3} + x^{2} \log{\left(x \right)}^{3}} - \frac{4}{- x^{4} \log{\left(x \right)}^{4} + 3 x^{3} \log{\left(x \right)}^{4} - 3 x^{2} \log{\left(x \right)}^{4} + x \log{\left(x \right)}^{4}} - \frac{2}{- x^{4} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(x \right)}^{3} - 3 x^{2} \log{\left(x \right)}^{3} + x \log{\left(x \right)}^{3}} + \frac{2}{- x^{3} \log{\left(x \right)}^{4} + 3 x^{2} \log{\left(x \right)}^{4} - 3 x \log{\left(x \right)}^{4} + \log{\left(x \right)}^{4}}}{- \frac{2}{x^{3} \log{\left(x \right)} - 3 x^{2} \log{\left(x \right)} + 3 x \log{\left(x \right)} - \log{\left(x \right)}} - \frac{2}{x^{3} \log{\left(x \right)}^{2} - 2 x^{2} \log{\left(x \right)}^{2} + x \log{\left(x \right)}^{2}} - \frac{2}{x^{3} \log{\left(x \right)}^{3} - x^{2} \log{\left(x \right)}^{3}} - \frac{1}{x^{3} \log{\left(x \right)}^{2} - x^{2} \log{\left(x \right)}^{2}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right) = 0$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ 1 1 \ lim |------ - ------| x->0+\log(x) -1 + x/
$$\lim_{x \to 0^+}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right)$$
1
$$1$$
= 0.885306813471745
/ 1 1 \ lim |------ - ------| x->0-\log(x) -1 + x/
$$\lim_{x \to 0^-}\left(\frac{1}{\log{\left(x \right)}} - \frac{1}{x - 1}\right)$$
1
$$1$$
= (0.898041579324335 - 0.0373313424703771j)
= (0.898041579324335 - 0.0373313424703771j)
Gráfico