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Límite de (-3+x^2-2*x)/(-15-4*x+3*x^2)
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Límite de (sqrt(5+x)-sqrt(10))/(-15+x^2-2*x)
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Límite de (sqrt(1+x+x^2)-sqrt(1+x^2-x))/(x^2-x)
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Expresiones idénticas
- x*(uno -(uno +x)*log(uno + dos /(uno +x)^ tres)/(x*log(uno + dos /x^ tres)))
- x multiplicar por (1 menos (1 más x) multiplicar por logaritmo de (1 más 2 dividir por (1 más x) al cubo ) dividir por (x multiplicar por logaritmo de (1 más 2 dividir por x al cubo )))
- x multiplicar por (uno menos (uno más x) multiplicar por logaritmo de (uno más dos dividir por (uno más x) en el grado tres) dividir por (x multiplicar por logaritmo de (uno más dos dividir por x en el grado tres)))
- x*(1-(1+x)*log(1+2/(1+x)3)/(x*log(1+2/x3)))
- x*1-1+x*log1+2/1+x3/x*log1+2/x3
- x*(1-(1+x)*log(1+2/(1+x)³)/(x*log(1+2/x³)))
- x*(1-(1+x)*log(1+2/(1+x) en el grado 3)/(x*log(1+2/x en el grado 3)))
- x(1-(1+x)log(1+2/(1+x)^3)/(xlog(1+2/x^3)))
- x(1-(1+x)log(1+2/(1+x)3)/(xlog(1+2/x3)))
- x1-1+xlog1+2/1+x3/xlog1+2/x3
- x1-1+xlog1+2/1+x^3/xlog1+2/x^3
- x*(1-(1+x)*log(1+2 dividir por (1+x)^3) dividir por (x*log(1+2 dividir por x^3)))
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Expresiones semejantes
- x*(1+(1+x)*log(1+2/(1+x)^3)/(x*log(1+2/x^3)))
- x*(1-(1+x)*log(1+2/(1-x)^3)/(x*log(1+2/x^3)))
- x*(1-(1+x)*log(1+2/(1+x)^3)/(x*log(1-2/x^3)))
- x*(1-(1+x)*log(1-2/(1+x)^3)/(x*log(1+2/x^3)))
- x*(1-(1-x)*log(1+2/(1+x)^3)/(x*log(1+2/x^3)))
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Expresiones con funciones
- Logaritmo log
- log(1+2*x)/(3*x+4*x^2)
- log(1+1/sqrt(x))
- log(5+x^2-4*x)/x
- log(1+t)/t
- log(1+6*x^2)/log(sqrt(1+4*sin(x)^2))
- Logaritmo log
- log(1+2*x)/(3*x+4*x^2)
- log(1+1/sqrt(x))
- log(5+x^2-4*x)/x
- log(1+t)/t
- log(1+6*x^2)/log(sqrt(1+4*sin(x)^2))
Límite de la función x*(1-(1+x)*log(1+2/(1+x)^3)/(x*log(1+2/x^3)))
Solución
Ha introducido
[src]
/ / / 2 \\\
| | (1 + x)*log|1 + --------|||
| | | 3|||
| | \ (1 + x) /||
lim |x*|1 - -------------------------||
x->oo| | / 2 \ ||
| | x*log|1 + --| ||
| | | 3| ||
\ \ \ x / //
$$\lim_{x \to \infty}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right)$$
Limit(x*(1 - (1 + x)*log(1 + 2/(1 + x)^3)/(x*log(1 + 2/x^3))), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{x^{3} + 2}{x^{3}} \right)}} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{x \log{\left(\frac{x^{3} + 2}{x^{3}} \right)} - \left(x + 1\right) \log{\left(\frac{\left(x + 1\right)^{3} + 2}{\left(x + 1\right)^{3}} \right)}} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x \log{\left(\frac{x^{3} + 2}{x^{3}} \right)} - \left(x + 1\right) \log{\left(\frac{\left(x + 1\right)^{3} + 2}{\left(x + 1\right)^{3}} \right)}}{\log{\left(\frac{x^{3} + 2}{x^{3}} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\frac{x^{3} + 2}{x^{3}} \right)}}}{\frac{d}{d x} \frac{1}{x \log{\left(\frac{x^{3} + 2}{x^{3}} \right)} - \left(x + 1\right) \log{\left(\frac{\left(x + 1\right)^{3} + 2}{\left(x + 1\right)^{3}} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(\frac{x^{4} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{6 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)} + \frac{x \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{3 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)}\right) \left(- \frac{3 x^{7}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{21 x^{6}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{63 x^{5}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{111 x^{4}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3 x^{4}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{129 x^{3}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x^{3}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{99 x^{2}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{18 x^{2}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{45 x}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \log{\left(1 + \frac{2}{x^{3}} \right)} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} - \frac{9}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} + \frac{6}{x^{3} + 2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(\frac{x^{4} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{6 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)} + \frac{x \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{3 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)}\right) \left(- \frac{3 x^{7}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{21 x^{6}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{63 x^{5}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{111 x^{4}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3 x^{4}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{129 x^{3}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x^{3}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{99 x^{2}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{18 x^{2}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{45 x}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \log{\left(1 + \frac{2}{x^{3}} \right)} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} - \frac{9}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} + \frac{6}{x^{3} + 2}\right)}\right)$$
=
$$2$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
oo/-oo,
tal que el límite para el numerador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{x^{3} + 2}{x^{3}} \right)}} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{x \log{\left(\frac{x^{3} + 2}{x^{3}} \right)} - \left(x + 1\right) \log{\left(\frac{\left(x + 1\right)^{3} + 2}{\left(x + 1\right)^{3}} \right)}} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x \log{\left(\frac{x^{3} + 2}{x^{3}} \right)} - \left(x + 1\right) \log{\left(\frac{\left(x + 1\right)^{3} + 2}{\left(x + 1\right)^{3}} \right)}}{\log{\left(\frac{x^{3} + 2}{x^{3}} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\frac{x^{3} + 2}{x^{3}} \right)}}}{\frac{d}{d x} \frac{1}{x \log{\left(\frac{x^{3} + 2}{x^{3}} \right)} - \left(x + 1\right) \log{\left(\frac{\left(x + 1\right)^{3} + 2}{\left(x + 1\right)^{3}} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(\frac{x^{4} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{6 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)} + \frac{x \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{3 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)}\right) \left(- \frac{3 x^{7}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{21 x^{6}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{63 x^{5}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{111 x^{4}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3 x^{4}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{129 x^{3}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x^{3}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{99 x^{2}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{18 x^{2}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{45 x}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \log{\left(1 + \frac{2}{x^{3}} \right)} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} - \frac{9}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} + \frac{6}{x^{3} + 2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(\frac{x^{4} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{6 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)} + \frac{x \log{\left(1 + \frac{2}{x^{3}} \right)}^{2}}{3 \left(x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)}^{2} - 2 x^{2} \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + x^{2} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} - 2 x \log{\left(1 + \frac{2}{x^{3}} \right)} \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} + 2 x \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)}^{2}\right)}\right) \left(- \frac{3 x^{7}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{21 x^{6}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{63 x^{5}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} - \frac{111 x^{4}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3 x^{4}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{129 x^{3}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x^{3}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{99 x^{2}}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{18 x^{2}}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \frac{45 x}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{12 x}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} - \log{\left(1 + \frac{2}{x^{3}} \right)} + \log{\left(\frac{x^{3}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x^{2}}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3 x}{x^{3} + 3 x^{2} + 3 x + 1} + \frac{3}{x^{3} + 3 x^{2} + 3 x + 1} \right)} - \frac{9}{x^{7} + 7 x^{6} + 21 x^{5} + 37 x^{4} + 43 x^{3} + 33 x^{2} + 15 x + 3} + \frac{3}{x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3} + \frac{6}{x^{3} + 2}\right)}\right)$$
=
$$2$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = 2$$
$$\lim_{x \to 0^-}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = \frac{- 2 \log{\left(5 \right)} + \log{\left(3 \right)} + 4 \log{\left(2 \right)}}{\log{\left(3 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = \frac{- 2 \log{\left(5 \right)} + \log{\left(3 \right)} + 4 \log{\left(2 \right)}}{\log{\left(3 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = 2$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = \frac{- 2 \log{\left(5 \right)} + \log{\left(3 \right)} + 4 \log{\left(2 \right)}}{\log{\left(3 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = \frac{- 2 \log{\left(5 \right)} + \log{\left(3 \right)} + 4 \log{\left(2 \right)}}{\log{\left(3 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \left(1 - \frac{\left(x + 1\right) \log{\left(1 + \frac{2}{\left(x + 1\right)^{3}} \right)}}{x \log{\left(1 + \frac{2}{x^{3}} \right)}}\right)\right) = 2$$
Más detalles con x→-oo