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- Límite de la función:
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Límite de (3+2*n)/(5+3*n)
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Límite de (1+4/x)^(2*x)
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Límite de (x/(-3+x))^(-5+x)
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Límite de ((4+3*x)/(-2+3*x))^(-7+5*x)
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Expresiones idénticas
- (sqrt(dos +x)-x)/(- ocho +x^ tres)
- ( raíz cuadrada de (2 más x) menos x) dividir por ( menos 8 más x al cubo )
- ( raíz cuadrada de (dos más x) menos x) dividir por ( menos ocho más x en el grado tres)
- (√(2+x)-x)/(-8+x^3)
- (sqrt(2+x)-x)/(-8+x3)
- sqrt2+x-x/-8+x3
- (sqrt(2+x)-x)/(-8+x³)
- (sqrt(2+x)-x)/(-8+x en el grado 3)
- sqrt2+x-x/-8+x^3
- (sqrt(2+x)-x) dividir por (-8+x^3)
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Expresiones semejantes
- (sqrt(2+x)-x)/(8+x^3)
- (sqrt(2+x)-x)/(-8-x^3)
- (sqrt(2-x)-x)/(-8+x^3)
- (sqrt(2+x)+x)/(-8+x^3)
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(-1+x^2)-sqrt(1+x^2)
- sqrt(3+2*x)-sqrt(-7+2*x)
- sqrt(x)*(sqrt(1+x)-sqrt(x))
- sqrt(n^2+2*n)-n
- sqrt(n)*(1+(1+n)^2)/(sqrt(1+n)*(1+n^2))
Límite de la función (sqrt(2+x)-x)/(-8+x^3)
Solución
Ha introducido
[src]
/ _______ \
|\/ 2 + x - x|
lim |-------------|
x->oo| 3 |
\ -8 + x /
$$\lim_{x \to \infty}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right)$$
Limit((sqrt(2 + x) - x)/(-8 + x^3), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(- x + \sqrt{x + 2}\right) = -\infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(x^{3} - 8\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- x + \sqrt{x + 2}\right)}{\frac{d}{d x} \left(x^{3} - 8\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{1}{2 \sqrt{x + 2}}}{3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(-1 + \frac{1}{2 \sqrt{x + 2}}\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{1}{24 x \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{x \sqrt{x + 2} + 2 \sqrt{x + 2}}}{\frac{d}{d x} \left(- 24 x\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{x}{2 \sqrt{x + 2}} - \sqrt{x + 2} - \frac{1}{\sqrt{x + 2}}}{- 24 x^{3} - 144 x^{2} - 288 x - 192}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{x}{2 \sqrt{x + 2}} - \sqrt{x + 2} - \frac{1}{\sqrt{x + 2}}\right)}{\frac{d}{d x} \left(- 24 x^{3} - 144 x^{2} - 288 x - 192\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} + \frac{1}{2 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} - \frac{1}{\sqrt{x + 2}}}{- 72 x^{2} - 288 x - 288}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{x}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} + \frac{1}{2 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} - \frac{1}{\sqrt{x + 2}}\right)}{\frac{d}{d x} \left(- 72 x^{2} - 288 x - 288\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{x^{2}}{8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{4 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{x}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} - \frac{\sqrt{x + 2}}{2 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} + \frac{3}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)}}{- 144 x - 288}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{x^{2}}{8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{4 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{x}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} - \frac{\sqrt{x + 2}}{2 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} + \frac{3}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)}\right)}{\frac{d}{d x} \left(- 144 x - 288\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{5}}{2304 x^{7} \sqrt{x + 2} + 32256 x^{6} \sqrt{x + 2} + 193536 x^{5} \sqrt{x + 2} + 645120 x^{4} \sqrt{x + 2} + 1290240 x^{3} \sqrt{x + 2} + 1548288 x^{2} \sqrt{x + 2} + 1032192 x \sqrt{x + 2} + 294912 \sqrt{x + 2}} - \frac{x^{4} \sqrt{x + 2}}{6 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{4}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{4}}{384 x^{7} \sqrt{x + 2} + 5376 x^{6} \sqrt{x + 2} + 32256 x^{5} \sqrt{x + 2} + 107520 x^{4} \sqrt{x + 2} + 215040 x^{3} \sqrt{x + 2} + 258048 x^{2} \sqrt{x + 2} + 172032 x \sqrt{x + 2} + 49152 \sqrt{x + 2}} - \frac{2 x^{3} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{3} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{3} \sqrt{x + 2}}{192 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{7 x^{3}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{3}}{192 x^{7} \sqrt{x + 2} + 2688 x^{6} \sqrt{x + 2} + 16128 x^{5} \sqrt{x + 2} + 53760 x^{4} \sqrt{x + 2} + 107520 x^{3} \sqrt{x + 2} + 129024 x^{2} \sqrt{x + 2} + 86016 x \sqrt{x + 2} + 24576 \sqrt{x + 2}} - \frac{2 x^{2} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{5 x^{2} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{2} \sqrt{x + 2}}{32 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{2 x^{2}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{2}}{32 x^{7} \sqrt{x + 2} + 448 x^{6} \sqrt{x + 2} + 2688 x^{5} \sqrt{x + 2} + 8960 x^{4} \sqrt{x + 2} + 17920 x^{3} \sqrt{x + 2} + 21504 x^{2} \sqrt{x + 2} + 14336 x \sqrt{x + 2} + 4096 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{3 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x \sqrt{x + 2}}{16 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{5 x}{144 x^{7} \sqrt{x + 2} + 2016 x^{6} \sqrt{x + 2} + 12096 x^{5} \sqrt{x + 2} + 40320 x^{4} \sqrt{x + 2} + 80640 x^{3} \sqrt{x + 2} + 96768 x^{2} \sqrt{x + 2} + 64512 x \sqrt{x + 2} + 18432 \sqrt{x + 2}} + \frac{x}{1152 x^{3} \sqrt{x + 2} + 6912 x^{2} \sqrt{x + 2} + 13824 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{x}{384 x^{3} \sqrt{x + 2} + 2304 x^{2} \sqrt{x + 2} + 4608 x \sqrt{x + 2} + 3072 \sqrt{x + 2}} + \frac{x}{72 \left(8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}\right)} - \frac{\sqrt{x + 2}}{6 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{\sqrt{x + 2}}{24 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} + \frac{\sqrt{x + 2}}{144 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{72 x^{7} \sqrt{x + 2} + 1008 x^{6} \sqrt{x + 2} + 6048 x^{5} \sqrt{x + 2} + 20160 x^{4} \sqrt{x + 2} + 40320 x^{3} \sqrt{x + 2} + 48384 x^{2} \sqrt{x + 2} + 32256 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{1}{576 x^{3} \sqrt{x + 2} + 3456 x^{2} \sqrt{x + 2} + 6912 x \sqrt{x + 2} + 4608 \sqrt{x + 2}} + \frac{1}{192 x^{3} \sqrt{x + 2} + 1152 x^{2} \sqrt{x + 2} + 2304 x \sqrt{x + 2} + 1536 \sqrt{x + 2}} + \frac{1}{144 \left(2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{5}}{2304 x^{7} \sqrt{x + 2} + 32256 x^{6} \sqrt{x + 2} + 193536 x^{5} \sqrt{x + 2} + 645120 x^{4} \sqrt{x + 2} + 1290240 x^{3} \sqrt{x + 2} + 1548288 x^{2} \sqrt{x + 2} + 1032192 x \sqrt{x + 2} + 294912 \sqrt{x + 2}} - \frac{x^{4} \sqrt{x + 2}}{6 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{4}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{4}}{384 x^{7} \sqrt{x + 2} + 5376 x^{6} \sqrt{x + 2} + 32256 x^{5} \sqrt{x + 2} + 107520 x^{4} \sqrt{x + 2} + 215040 x^{3} \sqrt{x + 2} + 258048 x^{2} \sqrt{x + 2} + 172032 x \sqrt{x + 2} + 49152 \sqrt{x + 2}} - \frac{2 x^{3} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{3} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{3} \sqrt{x + 2}}{192 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{7 x^{3}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{3}}{192 x^{7} \sqrt{x + 2} + 2688 x^{6} \sqrt{x + 2} + 16128 x^{5} \sqrt{x + 2} + 53760 x^{4} \sqrt{x + 2} + 107520 x^{3} \sqrt{x + 2} + 129024 x^{2} \sqrt{x + 2} + 86016 x \sqrt{x + 2} + 24576 \sqrt{x + 2}} - \frac{2 x^{2} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{5 x^{2} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{2} \sqrt{x + 2}}{32 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{2 x^{2}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{2}}{32 x^{7} \sqrt{x + 2} + 448 x^{6} \sqrt{x + 2} + 2688 x^{5} \sqrt{x + 2} + 8960 x^{4} \sqrt{x + 2} + 17920 x^{3} \sqrt{x + 2} + 21504 x^{2} \sqrt{x + 2} + 14336 x \sqrt{x + 2} + 4096 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{3 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x \sqrt{x + 2}}{16 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{5 x}{144 x^{7} \sqrt{x + 2} + 2016 x^{6} \sqrt{x + 2} + 12096 x^{5} \sqrt{x + 2} + 40320 x^{4} \sqrt{x + 2} + 80640 x^{3} \sqrt{x + 2} + 96768 x^{2} \sqrt{x + 2} + 64512 x \sqrt{x + 2} + 18432 \sqrt{x + 2}} + \frac{x}{1152 x^{3} \sqrt{x + 2} + 6912 x^{2} \sqrt{x + 2} + 13824 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{x}{384 x^{3} \sqrt{x + 2} + 2304 x^{2} \sqrt{x + 2} + 4608 x \sqrt{x + 2} + 3072 \sqrt{x + 2}} + \frac{x}{72 \left(8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}\right)} - \frac{\sqrt{x + 2}}{6 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{\sqrt{x + 2}}{24 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} + \frac{\sqrt{x + 2}}{144 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{72 x^{7} \sqrt{x + 2} + 1008 x^{6} \sqrt{x + 2} + 6048 x^{5} \sqrt{x + 2} + 20160 x^{4} \sqrt{x + 2} + 40320 x^{3} \sqrt{x + 2} + 48384 x^{2} \sqrt{x + 2} + 32256 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{1}{576 x^{3} \sqrt{x + 2} + 3456 x^{2} \sqrt{x + 2} + 6912 x \sqrt{x + 2} + 4608 \sqrt{x + 2}} + \frac{1}{192 x^{3} \sqrt{x + 2} + 1152 x^{2} \sqrt{x + 2} + 2304 x \sqrt{x + 2} + 1536 \sqrt{x + 2}} + \frac{1}{144 \left(2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}\right)}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 6 vez (veces)
-oo/oo,
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(- x + \sqrt{x + 2}\right) = -\infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(x^{3} - 8\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- x + \sqrt{x + 2}\right)}{\frac{d}{d x} \left(x^{3} - 8\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{1}{2 \sqrt{x + 2}}}{3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(-1 + \frac{1}{2 \sqrt{x + 2}}\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{1}{24 x \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{x \sqrt{x + 2} + 2 \sqrt{x + 2}}}{\frac{d}{d x} \left(- 24 x\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{x}{2 \sqrt{x + 2}} - \sqrt{x + 2} - \frac{1}{\sqrt{x + 2}}}{- 24 x^{3} - 144 x^{2} - 288 x - 192}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{x}{2 \sqrt{x + 2}} - \sqrt{x + 2} - \frac{1}{\sqrt{x + 2}}\right)}{\frac{d}{d x} \left(- 24 x^{3} - 144 x^{2} - 288 x - 192\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} + \frac{1}{2 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} - \frac{1}{\sqrt{x + 2}}}{- 72 x^{2} - 288 x - 288}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{x}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} + \frac{1}{2 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)} - \frac{1}{\sqrt{x + 2}}\right)}{\frac{d}{d x} \left(- 72 x^{2} - 288 x - 288\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{x^{2}}{8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{4 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{x}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} - \frac{\sqrt{x + 2}}{2 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} + \frac{3}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)}}{- 144 x - 288}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{x^{2}}{8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{4 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{x}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} - \frac{\sqrt{x + 2}}{2 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}} + \frac{3}{4 \left(x \sqrt{x + 2} + 2 \sqrt{x + 2}\right)}\right)}{\frac{d}{d x} \left(- 144 x - 288\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{5}}{2304 x^{7} \sqrt{x + 2} + 32256 x^{6} \sqrt{x + 2} + 193536 x^{5} \sqrt{x + 2} + 645120 x^{4} \sqrt{x + 2} + 1290240 x^{3} \sqrt{x + 2} + 1548288 x^{2} \sqrt{x + 2} + 1032192 x \sqrt{x + 2} + 294912 \sqrt{x + 2}} - \frac{x^{4} \sqrt{x + 2}}{6 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{4}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{4}}{384 x^{7} \sqrt{x + 2} + 5376 x^{6} \sqrt{x + 2} + 32256 x^{5} \sqrt{x + 2} + 107520 x^{4} \sqrt{x + 2} + 215040 x^{3} \sqrt{x + 2} + 258048 x^{2} \sqrt{x + 2} + 172032 x \sqrt{x + 2} + 49152 \sqrt{x + 2}} - \frac{2 x^{3} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{3} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{3} \sqrt{x + 2}}{192 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{7 x^{3}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{3}}{192 x^{7} \sqrt{x + 2} + 2688 x^{6} \sqrt{x + 2} + 16128 x^{5} \sqrt{x + 2} + 53760 x^{4} \sqrt{x + 2} + 107520 x^{3} \sqrt{x + 2} + 129024 x^{2} \sqrt{x + 2} + 86016 x \sqrt{x + 2} + 24576 \sqrt{x + 2}} - \frac{2 x^{2} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{5 x^{2} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{2} \sqrt{x + 2}}{32 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{2 x^{2}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{2}}{32 x^{7} \sqrt{x + 2} + 448 x^{6} \sqrt{x + 2} + 2688 x^{5} \sqrt{x + 2} + 8960 x^{4} \sqrt{x + 2} + 17920 x^{3} \sqrt{x + 2} + 21504 x^{2} \sqrt{x + 2} + 14336 x \sqrt{x + 2} + 4096 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{3 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x \sqrt{x + 2}}{16 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{5 x}{144 x^{7} \sqrt{x + 2} + 2016 x^{6} \sqrt{x + 2} + 12096 x^{5} \sqrt{x + 2} + 40320 x^{4} \sqrt{x + 2} + 80640 x^{3} \sqrt{x + 2} + 96768 x^{2} \sqrt{x + 2} + 64512 x \sqrt{x + 2} + 18432 \sqrt{x + 2}} + \frac{x}{1152 x^{3} \sqrt{x + 2} + 6912 x^{2} \sqrt{x + 2} + 13824 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{x}{384 x^{3} \sqrt{x + 2} + 2304 x^{2} \sqrt{x + 2} + 4608 x \sqrt{x + 2} + 3072 \sqrt{x + 2}} + \frac{x}{72 \left(8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}\right)} - \frac{\sqrt{x + 2}}{6 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{\sqrt{x + 2}}{24 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} + \frac{\sqrt{x + 2}}{144 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{72 x^{7} \sqrt{x + 2} + 1008 x^{6} \sqrt{x + 2} + 6048 x^{5} \sqrt{x + 2} + 20160 x^{4} \sqrt{x + 2} + 40320 x^{3} \sqrt{x + 2} + 48384 x^{2} \sqrt{x + 2} + 32256 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{1}{576 x^{3} \sqrt{x + 2} + 3456 x^{2} \sqrt{x + 2} + 6912 x \sqrt{x + 2} + 4608 \sqrt{x + 2}} + \frac{1}{192 x^{3} \sqrt{x + 2} + 1152 x^{2} \sqrt{x + 2} + 2304 x \sqrt{x + 2} + 1536 \sqrt{x + 2}} + \frac{1}{144 \left(2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{5}}{2304 x^{7} \sqrt{x + 2} + 32256 x^{6} \sqrt{x + 2} + 193536 x^{5} \sqrt{x + 2} + 645120 x^{4} \sqrt{x + 2} + 1290240 x^{3} \sqrt{x + 2} + 1548288 x^{2} \sqrt{x + 2} + 1032192 x \sqrt{x + 2} + 294912 \sqrt{x + 2}} - \frac{x^{4} \sqrt{x + 2}}{6 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{4}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{4}}{384 x^{7} \sqrt{x + 2} + 5376 x^{6} \sqrt{x + 2} + 32256 x^{5} \sqrt{x + 2} + 107520 x^{4} \sqrt{x + 2} + 215040 x^{3} \sqrt{x + 2} + 258048 x^{2} \sqrt{x + 2} + 172032 x \sqrt{x + 2} + 49152 \sqrt{x + 2}} - \frac{2 x^{3} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{x^{3} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{3} \sqrt{x + 2}}{192 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{7 x^{3}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{3}}{192 x^{7} \sqrt{x + 2} + 2688 x^{6} \sqrt{x + 2} + 16128 x^{5} \sqrt{x + 2} + 53760 x^{4} \sqrt{x + 2} + 107520 x^{3} \sqrt{x + 2} + 129024 x^{2} \sqrt{x + 2} + 86016 x \sqrt{x + 2} + 24576 \sqrt{x + 2}} - \frac{2 x^{2} \sqrt{x + 2}}{3 \left(64 x^{7} + 896 x^{6} + 5376 x^{5} + 17920 x^{4} + 35840 x^{3} + 43008 x^{2} + 28672 x + 8192\right)} - \frac{5 x^{2} \sqrt{x + 2}}{24 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x^{2} \sqrt{x + 2}}{32 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{2 x^{2}}{576 x^{7} \sqrt{x + 2} + 8064 x^{6} \sqrt{x + 2} + 48384 x^{5} \sqrt{x + 2} + 161280 x^{4} \sqrt{x + 2} + 322560 x^{3} \sqrt{x + 2} + 387072 x^{2} \sqrt{x + 2} + 258048 x \sqrt{x + 2} + 73728 \sqrt{x + 2}} - \frac{x^{2}}{32 x^{7} \sqrt{x + 2} + 448 x^{6} \sqrt{x + 2} + 2688 x^{5} \sqrt{x + 2} + 8960 x^{4} \sqrt{x + 2} + 17920 x^{3} \sqrt{x + 2} + 21504 x^{2} \sqrt{x + 2} + 14336 x \sqrt{x + 2} + 4096 \sqrt{x + 2}} - \frac{x \sqrt{x + 2}}{3 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{x \sqrt{x + 2}}{16 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} - \frac{5 x}{144 x^{7} \sqrt{x + 2} + 2016 x^{6} \sqrt{x + 2} + 12096 x^{5} \sqrt{x + 2} + 40320 x^{4} \sqrt{x + 2} + 80640 x^{3} \sqrt{x + 2} + 96768 x^{2} \sqrt{x + 2} + 64512 x \sqrt{x + 2} + 18432 \sqrt{x + 2}} + \frac{x}{1152 x^{3} \sqrt{x + 2} + 6912 x^{2} \sqrt{x + 2} + 13824 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{x}{384 x^{3} \sqrt{x + 2} + 2304 x^{2} \sqrt{x + 2} + 4608 x \sqrt{x + 2} + 3072 \sqrt{x + 2}} + \frac{x}{72 \left(8 x^{3} \sqrt{x + 2} + 48 x^{2} \sqrt{x + 2} + 96 x \sqrt{x + 2} + 64 \sqrt{x + 2}\right)} - \frac{\sqrt{x + 2}}{6 \left(4 x^{7} + 56 x^{6} + 336 x^{5} + 1120 x^{4} + 2240 x^{3} + 2688 x^{2} + 1792 x + 512\right)} - \frac{\sqrt{x + 2}}{24 \left(x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64\right)} + \frac{\sqrt{x + 2}}{144 \left(x^{3} + 6 x^{2} + 12 x + 8\right)} - \frac{1}{72 x^{7} \sqrt{x + 2} + 1008 x^{6} \sqrt{x + 2} + 6048 x^{5} \sqrt{x + 2} + 20160 x^{4} \sqrt{x + 2} + 40320 x^{3} \sqrt{x + 2} + 48384 x^{2} \sqrt{x + 2} + 32256 x \sqrt{x + 2} + 9216 \sqrt{x + 2}} + \frac{1}{576 x^{3} \sqrt{x + 2} + 3456 x^{2} \sqrt{x + 2} + 6912 x \sqrt{x + 2} + 4608 \sqrt{x + 2}} + \frac{1}{192 x^{3} \sqrt{x + 2} + 1152 x^{2} \sqrt{x + 2} + 2304 x \sqrt{x + 2} + 1536 \sqrt{x + 2}} + \frac{1}{144 \left(2 x^{3} \sqrt{x + 2} + 12 x^{2} \sqrt{x + 2} + 24 x \sqrt{x + 2} + 16 \sqrt{x + 2}\right)}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 6 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = - \frac{\sqrt{2}}{8}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = - \frac{\sqrt{2}}{8}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = \frac{1}{7} - \frac{\sqrt{3}}{7}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = \frac{1}{7} - \frac{\sqrt{3}}{7}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = 0$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = - \frac{\sqrt{2}}{8}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = - \frac{\sqrt{2}}{8}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = \frac{1}{7} - \frac{\sqrt{3}}{7}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = \frac{1}{7} - \frac{\sqrt{3}}{7}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x + \sqrt{x + 2}}{x^{3} - 8}\right) = 0$$
Más detalles con x→-oo