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Expresiones semejantes
- x^4*log(1-2/x)/64
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Expresiones con funciones
- Logaritmo log
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- log(sin(x))*sin(x)
Límite de la función x^4*log(1+2/x)/64
Solución
Ha introducido
[src]
/ 4 / 2\\
|x *log|1 + -||
| \ x/|
lim |-------------|
x->oo\ 64 /
$$\lim_{x \to \infty}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right)$$
Limit((x^4*log(1 + 2/x))/64, x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(\frac{x^{4}}{64}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{x + 2}{x} \right)}} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x^{4} \log{\left(\frac{x + 2}{x} \right)}}{64}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{x^{4}}{64}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{x + 2}{x} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{2} \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{16 \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{x^{2} \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{16}\right)}{\frac{d}{d x} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} - \frac{x^{2} \log{\left(\frac{x + 2}{x} \right)}^{2}}{16} - \frac{x \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}}{- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}}}{\frac{d}{d x} \frac{1}{- \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} - \frac{x^{2} \log{\left(\frac{x + 2}{x} \right)}^{2}}{16} - \frac{x \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}\right) \left(- \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} - \frac{x^{2} \log{\left(\frac{x + 2}{x} \right)}^{2}}{16} - \frac{x \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}\right)^{2}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2} \left(\frac{x^{4} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{8 \left(x + 2\right)} + \frac{x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} + \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8 \left(x + 2\right)} + \frac{5 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} + \frac{x \log{\left(\frac{x + 2}{x} \right)}^{2}}{4} + \frac{\left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2} \left(\frac{2 x^{4} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} + 2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 10 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 4 x \log{\left(\frac{x + 2}{x} \right)}^{2} + 2 \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2}}}{\frac{d}{d x} \left(\frac{2 x^{4} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} + 2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 10 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 4 x \log{\left(\frac{x + 2}{x} \right)}^{2} + 2 \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\frac{24}{x^{4}} - \frac{24 \left(x + 2\right)}{x^{5}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2}} + \frac{\left(- \frac{12}{x^{3}} + \frac{12 \left(x + 2\right)}{x^{4}}\right) \left(- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}\right)}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{3}}}{\frac{2 x^{4} \left(- \frac{4}{x^{2}} + \frac{4 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + \frac{2 x^{4} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + 2 x^{3} \left(\frac{6}{x^{3}} - \frac{6 \left(x + 2\right)}{x^{4}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + \frac{18 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} - \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{\left(x + 2\right)^{2}} + 16 x^{2} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{14 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 24 x \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 6 \log{\left(\frac{x + 2}{x} \right)}^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\frac{24}{x^{4}} - \frac{24 \left(x + 2\right)}{x^{5}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2}} + \frac{\left(- \frac{12}{x^{3}} + \frac{12 \left(x + 2\right)}{x^{4}}\right) \left(- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}\right)}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{3}}\right)}{\frac{d}{d x} \left(\frac{2 x^{4} \left(- \frac{4}{x^{2}} + \frac{4 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + \frac{2 x^{4} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + 2 x^{3} \left(\frac{6}{x^{3}} - \frac{6 \left(x + 2\right)}{x^{4}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + \frac{18 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} - \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{\left(x + 2\right)^{2}} + 16 x^{2} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{14 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 24 x \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 6 \log{\left(\frac{x + 2}{x} \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3}{2 \left(- \frac{8 x \log{\left(1 + \frac{2}{x} \right)}}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{4 x}{x^{2} + 4 x + 4} - \frac{8}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{32}{x^{3} + 4 x^{2} + 4 x} + \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 4 x + 4} + \frac{16}{x^{2} + 4 x + 4} - \frac{352}{x^{3} + 2 x^{2}} - \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 2 x} - \frac{344}{x^{2} + 2 x} - \frac{86}{x + 2} + \frac{82}{x} + \frac{8 \log{\left(1 + \frac{2}{x} \right)}}{x^{2}} + \frac{172}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3}{2 \left(- \frac{8 x \log{\left(1 + \frac{2}{x} \right)}}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{4 x}{x^{2} + 4 x + 4} - \frac{8}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{32}{x^{3} + 4 x^{2} + 4 x} + \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 4 x + 4} + \frac{16}{x^{2} + 4 x + 4} - \frac{352}{x^{3} + 2 x^{2}} - \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 2 x} - \frac{344}{x^{2} + 2 x} - \frac{86}{x + 2} + \frac{82}{x} + \frac{8 \log{\left(1 + \frac{2}{x} \right)}}{x^{2}} + \frac{172}{x^{2}}\right)}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
oo/oo,
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(\frac{x^{4}}{64}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{x + 2}{x} \right)}} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x^{4} \log{\left(\frac{x + 2}{x} \right)}}{64}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{x^{4}}{64}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{x + 2}{x} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{2} \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{16 \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{x^{2} \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{16}\right)}{\frac{d}{d x} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} - \frac{x^{2} \log{\left(\frac{x + 2}{x} \right)}^{2}}{16} - \frac{x \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}}{- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}}}{\frac{d}{d x} \frac{1}{- \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} - \frac{x^{2} \log{\left(\frac{x + 2}{x} \right)}^{2}}{16} - \frac{x \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}\right) \left(- \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} - \frac{x^{2} \log{\left(\frac{x + 2}{x} \right)}^{2}}{16} - \frac{x \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}\right)^{2}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2} \left(\frac{x^{4} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{8 \left(x + 2\right)} + \frac{x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} + \frac{x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8 \left(x + 2\right)} + \frac{5 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{8} + \frac{x \log{\left(\frac{x + 2}{x} \right)}^{2}}{4} + \frac{\left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}}{8}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2} \left(\frac{2 x^{4} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} + 2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 10 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 4 x \log{\left(\frac{x + 2}{x} \right)}^{2} + 2 \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2}}}{\frac{d}{d x} \left(\frac{2 x^{4} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} + 2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 10 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 4 x \log{\left(\frac{x + 2}{x} \right)}^{2} + 2 \left(x + 2\right) \log{\left(\frac{x + 2}{x} \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\frac{24}{x^{4}} - \frac{24 \left(x + 2\right)}{x^{5}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2}} + \frac{\left(- \frac{12}{x^{3}} + \frac{12 \left(x + 2\right)}{x^{4}}\right) \left(- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}\right)}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{3}}}{\frac{2 x^{4} \left(- \frac{4}{x^{2}} + \frac{4 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + \frac{2 x^{4} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + 2 x^{3} \left(\frac{6}{x^{3}} - \frac{6 \left(x + 2\right)}{x^{4}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + \frac{18 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} - \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{\left(x + 2\right)^{2}} + 16 x^{2} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{14 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 24 x \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 6 \log{\left(\frac{x + 2}{x} \right)}^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\frac{24}{x^{4}} - \frac{24 \left(x + 2\right)}{x^{5}}}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{2}} + \frac{\left(- \frac{12}{x^{3}} + \frac{12 \left(x + 2\right)}{x^{4}}\right) \left(- \frac{6}{x^{3}} + \frac{6 \left(x + 2\right)}{x^{4}}\right)}{\left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right)^{3}}\right)}{\frac{d}{d x} \left(\frac{2 x^{4} \left(- \frac{4}{x^{2}} + \frac{4 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + \frac{2 x^{4} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)}{x + 2} + 2 x^{3} \left(\frac{6}{x^{3}} - \frac{6 \left(x + 2\right)}{x^{4}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{2 x^{3} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + \frac{18 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right)^{2}}{x + 2} - \frac{2 x^{3} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{\left(x + 2\right)^{2}} + 16 x^{2} \left(- \frac{2}{x^{2}} + \frac{2 \left(x + 2\right)}{x^{3}}\right) \log{\left(\frac{x + 2}{x} \right)} + \frac{14 x^{2} \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)}}{x + 2} + 24 x \left(\frac{1}{x} - \frac{x + 2}{x^{2}}\right) \log{\left(\frac{x + 2}{x} \right)} + 6 \log{\left(\frac{x + 2}{x} \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3}{2 \left(- \frac{8 x \log{\left(1 + \frac{2}{x} \right)}}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{4 x}{x^{2} + 4 x + 4} - \frac{8}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{32}{x^{3} + 4 x^{2} + 4 x} + \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 4 x + 4} + \frac{16}{x^{2} + 4 x + 4} - \frac{352}{x^{3} + 2 x^{2}} - \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 2 x} - \frac{344}{x^{2} + 2 x} - \frac{86}{x + 2} + \frac{82}{x} + \frac{8 \log{\left(1 + \frac{2}{x} \right)}}{x^{2}} + \frac{172}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3}{2 \left(- \frac{8 x \log{\left(1 + \frac{2}{x} \right)}}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{4 x}{x^{2} + 4 x + 4} - \frac{8}{x^{3} + 6 x^{2} + 12 x + 8} + \frac{32}{x^{3} + 4 x^{2} + 4 x} + \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 4 x + 4} + \frac{16}{x^{2} + 4 x + 4} - \frac{352}{x^{3} + 2 x^{2}} - \frac{24 \log{\left(1 + \frac{2}{x} \right)}}{x^{2} + 2 x} - \frac{344}{x^{2} + 2 x} - \frac{86}{x + 2} + \frac{82}{x} + \frac{8 \log{\left(1 + \frac{2}{x} \right)}}{x^{2}} + \frac{172}{x^{2}}\right)}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = \frac{\log{\left(3 \right)}}{64}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = \frac{\log{\left(3 \right)}}{64}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = -\infty$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = \frac{\log{\left(3 \right)}}{64}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = \frac{\log{\left(3 \right)}}{64}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x^{4} \log{\left(1 + \frac{2}{x} \right)}}{64}\right) = -\infty$$
Más detalles con x→-oo