Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
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Límite de sin(x)/x
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Límite de tan(x)/x
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Límite de sin(3*x)/x
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Límite de x^sin(x)
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Expresiones idénticas
- sqrt(dos +x)*sqrt(sin((uno +x)/(uno +(uno +x)^ tres)))/(sqrt(uno +x)*sqrt(sin(x/(uno +x^ tres))))
- raíz cuadrada de (2 más x) multiplicar por raíz cuadrada de ( seno de ((1 más x) dividir por (1 más (1 más x) al cubo ))) dividir por ( raíz cuadrada de (1 más x) multiplicar por raíz cuadrada de ( seno de (x dividir por (1 más x al cubo ))))
- raíz cuadrada de (dos más x) multiplicar por raíz cuadrada de ( seno de ((uno más x) dividir por (uno más (uno más x) en el grado tres))) dividir por ( raíz cuadrada de (uno más x) multiplicar por raíz cuadrada de ( seno de (x dividir por (uno más x en el grado tres))))
- √(2+x)*√(sin((1+x)/(1+(1+x)^3)))/(√(1+x)*√(sin(x/(1+x^3))))
- sqrt(2+x)*sqrt(sin((1+x)/(1+(1+x)3)))/(sqrt(1+x)*sqrt(sin(x/(1+x3))))
- sqrt2+x*sqrtsin1+x/1+1+x3/sqrt1+x*sqrtsinx/1+x3
- sqrt(2+x)*sqrt(sin((1+x)/(1+(1+x)³)))/(sqrt(1+x)*sqrt(sin(x/(1+x³))))
- sqrt(2+x)*sqrt(sin((1+x)/(1+(1+x) en el grado 3)))/(sqrt(1+x)*sqrt(sin(x/(1+x en el grado 3))))
- sqrt(2+x)sqrt(sin((1+x)/(1+(1+x)^3)))/(sqrt(1+x)sqrt(sin(x/(1+x^3))))
- sqrt(2+x)sqrt(sin((1+x)/(1+(1+x)3)))/(sqrt(1+x)sqrt(sin(x/(1+x3))))
- sqrt2+xsqrtsin1+x/1+1+x3/sqrt1+xsqrtsinx/1+x3
- sqrt2+xsqrtsin1+x/1+1+x^3/sqrt1+xsqrtsinx/1+x^3
- sqrt(2+x)*sqrt(sin((1+x) dividir por (1+(1+x)^3))) dividir por (sqrt(1+x)*sqrt(sin(x dividir por (1+x^3))))
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Expresiones semejantes
- sqrt(2-x)*sqrt(sin((1+x)/(1+(1+x)^3)))/(sqrt(1+x)*sqrt(sin(x/(1+x^3))))
- sqrt(2+x)*sqrt(sin((1+x)/(1+(1+x)^3)))/(sqrt(1-x)*sqrt(sin(x/(1+x^3))))
- sqrt(2+x)*sqrt(sin((1-x)/(1+(1+x)^3)))/(sqrt(1+x)*sqrt(sin(x/(1+x^3))))
- sqrt(2+x)*sqrt(sin((1+x)/(1+(1-x)^3)))/(sqrt(1+x)*sqrt(sin(x/(1+x^3))))
- sqrt(2+x)*sqrt(sin((1+x)/(1-(1+x)^3)))/(sqrt(1+x)*sqrt(sin(x/(1+x^3))))
- sqrt(2+x)*sqrt(sin((1+x)/(1+(1+x)^3)))/(sqrt(1+x)*sqrt(sin(x/(1-x^3))))
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(4+x^2)-x
- sqrt(n)
- sqrt(x+9*x^2)-3*x
- sqrt(3-x)/sqrt(1-x)
- sqrt(1-x)
- Raíz cuadrada sqrt
- sqrt(4+x^2)-x
- sqrt(n)
- sqrt(x+9*x^2)-3*x
- sqrt(3-x)/sqrt(1-x)
- sqrt(1-x)
- Seno sin
- sin(x)/x
- sin(3*x)/x
- sin(5*x)/(3*x)
- sin(5*x)/sin(3*x)
- sin(10*x)/x
- Raíz cuadrada sqrt
- sqrt(4+x^2)-x
- sqrt(n)
- sqrt(x+9*x^2)-3*x
- sqrt(3-x)/sqrt(1-x)
- sqrt(1-x)
- Raíz cuadrada sqrt
- sqrt(4+x^2)-x
- sqrt(n)
- sqrt(x+9*x^2)-3*x
- sqrt(3-x)/sqrt(1-x)
- sqrt(1-x)
- Seno sin
- sin(x)/x
- sin(3*x)/x
- sin(5*x)/(3*x)
- sin(5*x)/sin(3*x)
- sin(10*x)/x
Límite de la función sqrt(2+x)*sqrt(sin((1+x)/(1+(1+x)^3)))/(sqrt(1+x)*sqrt(sin(x/(1+x^3))))
Solución
Ha introducido
[src]
/ ___________________\
| _______ / / 1 + x \ |
|\/ 2 + x * / sin|------------| |
| / | 3| |
| \/ \1 + (1 + x) / |
lim |----------------------------------|
x->oo| _____________ |
| _______ / / x \ |
| \/ 1 + x * / sin|------| |
| / | 3| |
\ \/ \1 + x / /
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right)$$
Limit((sqrt(2 + x)*sqrt(sin((1 + x)/(1 + (1 + x)^3))))/((sqrt(1 + x)*sqrt(sin(x/(1 + x^3))))), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{\sqrt{x + 2}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}}{\frac{d}{d x} \frac{1}{\sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{2 \left(- \frac{\sqrt{x + 2} \left(- \frac{3 x^{3}}{\left(x^{3} + 1\right)^{2}} + \frac{1}{x^{3} + 1}\right) \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} + \frac{1}{2 \sqrt{x + 1} \sqrt{x + 2} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}} - \frac{\sqrt{x + 2}}{2 \left(x + 1\right)^{\frac{3}{2}} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) \sin^{\frac{3}{2}}{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}{\left(- \frac{3 \left(x + 1\right)^{3}}{\left(\left(x + 1\right)^{3} + 1\right)^{2}} + \frac{1}{\left(x + 1\right)^{3} + 1}\right) \cos{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{3 x^{3} \sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{6} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 4 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2}}{2 x \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}} + 2 \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}} + \frac{1}{2 \sqrt{x + 1} \sqrt{x + 2} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}}{\frac{3 x^{3}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x^{2}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{3}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} - \frac{1}{2 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 4 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{3 x^{3} \sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{6} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 4 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2}}{2 x \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}} + 2 \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}} + \frac{1}{2 \sqrt{x + 1} \sqrt{x + 2} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}}{\frac{3 x^{3}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x^{2}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{3}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} - \frac{1}{2 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 4 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}}}\right)$$
=
$$1$$
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
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{\sqrt{x + 2}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}}{\frac{d}{d x} \frac{1}{\sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{2 \left(- \frac{\sqrt{x + 2} \left(- \frac{3 x^{3}}{\left(x^{3} + 1\right)^{2}} + \frac{1}{x^{3} + 1}\right) \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} + \frac{1}{2 \sqrt{x + 1} \sqrt{x + 2} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}} - \frac{\sqrt{x + 2}}{2 \left(x + 1\right)^{\frac{3}{2}} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) \sin^{\frac{3}{2}}{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}{\left(- \frac{3 \left(x + 1\right)^{3}}{\left(\left(x + 1\right)^{3} + 1\right)^{2}} + \frac{1}{\left(x + 1\right)^{3} + 1}\right) \cos{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{3 x^{3} \sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{6} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 4 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2}}{2 x \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}} + 2 \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}} + \frac{1}{2 \sqrt{x + 1} \sqrt{x + 2} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}}{\frac{3 x^{3}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x^{2}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{3}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} - \frac{1}{2 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 4 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{3 x^{3} \sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{6} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 4 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2} \cos{\left(\frac{x}{x^{3} + 1} \right)}}{2 x^{3} \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)} + 2 \sqrt{x + 1} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 1} \right)}} - \frac{\sqrt{x + 2}}{2 x \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}} + 2 \sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}} + \frac{1}{2 \sqrt{x + 1} \sqrt{x + 2} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}}{\frac{3 x^{3}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x^{2}}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{9 x}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} + \frac{3}{2 x^{6} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 12 x^{5} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 30 x^{4} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 44 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 42 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 24 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 8 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}} - \frac{1}{2 x^{3} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x^{2} \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 6 x \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)} + 4 \sin^{\frac{3}{2}}{\left(\frac{x}{x^{3} + 3 x^{2} + 3 x + 2} + \frac{1}{x^{3} + 3 x^{2} + 3 x + 2} \right)}}}\right)$$
=
$$1$$
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
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = 1$$
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = - \infty i$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \frac{\sqrt{6} \sqrt{\sin{\left(\frac{2}{9} \right)}}}{2 \sqrt{\sin{\left(\frac{1}{2} \right)}}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \frac{\sqrt{6} \sqrt{\sin{\left(\frac{2}{9} \right)}}}{2 \sqrt{\sin{\left(\frac{1}{2} \right)}}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = 1$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = - \infty i$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \frac{\sqrt{6} \sqrt{\sin{\left(\frac{2}{9} \right)}}}{2 \sqrt{\sin{\left(\frac{1}{2} \right)}}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = \frac{\sqrt{6} \sqrt{\sin{\left(\frac{2}{9} \right)}}}{2 \sqrt{\sin{\left(\frac{1}{2} \right)}}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x + 2} \sqrt{\sin{\left(\frac{x + 1}{\left(x + 1\right)^{3} + 1} \right)}}}{\sqrt{x + 1} \sqrt{\sin{\left(\frac{x}{x^{3} + 1} \right)}}}\right) = 1$$
Más detalles con x→-oo