Sr Examen
Otras calculadoras:
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- Límite de la función:
-
Límite de (-1+x^3)/(-1+x)
-
Límite de (-9+x^2)/(-3+x)
-
Límite de (-3+x)/(-9+x^2)
-
Límite de x^sin(x)
-
Expresiones idénticas
- cot(x)^sin(x)*(cos(x)*log(cot(x))+(- uno -cot(x)^ dos)*sin(x)/cot(x))
- cotangente de (x) en el grado seno de (x) multiplicar por ( coseno de (x) multiplicar por logaritmo de ( cotangente de (x)) más ( menos 1 menos cotangente de (x) al cuadrado ) multiplicar por seno de (x) dividir por cotangente de (x))
- cotangente de (x) en el grado seno de (x) multiplicar por ( coseno de (x) multiplicar por logaritmo de ( cotangente de (x)) más ( menos uno menos cotangente de (x) en el grado dos) multiplicar por seno de (x) dividir por cotangente de (x))
- cot(x)sin(x)*(cos(x)*log(cot(x))+(-1-cot(x)2)*sin(x)/cot(x))
- cotxsinx*cosx*logcotx+-1-cotx2*sinx/cotx
- cot(x)^sin(x)*(cos(x)*log(cot(x))+(-1-cot(x)²)*sin(x)/cot(x))
- cot(x) en el grado sin(x)*(cos(x)*log(cot(x))+(-1-cot(x) en el grado 2)*sin(x)/cot(x))
- cot(x)^sin(x)(cos(x)log(cot(x))+(-1-cot(x)^2)sin(x)/cot(x))
- cot(x)sin(x)(cos(x)log(cot(x))+(-1-cot(x)2)sin(x)/cot(x))
- cotxsinxcosxlogcotx+-1-cotx2sinx/cotx
- cotx^sinxcosxlogcotx+-1-cotx^2sinx/cotx
- cot(x)^sin(x)*(cos(x)*log(cot(x))+(-1-cot(x)^2)*sin(x) dividir por cot(x))
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Expresiones semejantes
- cot(x)^sin(x)*(cos(x)*log(cot(x))+(1-cot(x)^2)*sin(x)/cot(x))
- cot(x)^sin(x)*(cos(x)*log(cot(x))+(-1+cot(x)^2)*sin(x)/cot(x))
- cot(x)^sin(x)*(cos(x)*log(cot(x))-(-1-cot(x)^2)*sin(x)/cot(x))
- cot(x)^sinx*(cosx*log(cot(x))+(-1-cot(x)^2)*sinx/cot(x))
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Expresiones con funciones
- Cotangente cot
- cot(2*x)*sin(3*x)
- cot(3*x)*sin(5*x)
- cot(3*x)/cot(6*x)
- cot(3*x)*sin(2*x)
- cot(5*x)*sin(7*x)
- Seno sin
- sin(7*x)/x
- sin(7*x)/(x^2+pi*x)
- sin(3*x)/(6*x)
- sin(6*x)
- sin(4*x)*tan(3*x)/(2*x^2)
- Coseno cos
- cos(x)^(1/sin(x))
- cos(2*x)/x
- cos(1/x)*sin(x)
- cos(pi*x/2)*log(1-x)
- cos(x)/x
- Logaritmo log
- log(x)/log(sin(x))
- log(-1+x)/cot(pi*x)
- log(1+x)/asin(x)
- log(x)/(-2+x+x^2)
- log(x/cot(x))
- Cotangente cot
- cot(2*x)*sin(3*x)
- cot(3*x)*sin(5*x)
- cot(3*x)/cot(6*x)
- cot(3*x)*sin(2*x)
- cot(5*x)*sin(7*x)
- Cotangente cot
- cot(2*x)*sin(3*x)
- cot(3*x)*sin(5*x)
- cot(3*x)/cot(6*x)
- cot(3*x)*sin(2*x)
- cot(5*x)*sin(7*x)
- Seno sin
- sin(7*x)/x
- sin(7*x)/(x^2+pi*x)
- sin(3*x)/(6*x)
- sin(6*x)
- sin(4*x)*tan(3*x)/(2*x^2)
- Cotangente cot
- cot(2*x)*sin(3*x)
- cot(3*x)*sin(5*x)
- cot(3*x)/cot(6*x)
- cot(3*x)*sin(2*x)
- cot(5*x)*sin(7*x)
Límite de la función cot(x)^sin(x)*(cos(x)*log(cot(x))+(-1-cot(x)^2)*sin(x)/cot(x))
Solución
Ha introducido
[src]
/ / / 2 \ \\
| sin(x) | \-1 - cot (x)/*sin(x)||
lim |cot (x)*|cos(x)*log(cot(x)) + ---------------------||
x->0+\ \ cot(x) //
$$\lim_{x \to 0^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
Limit(cot(x)^sin(x)*(cos(x)*log(cot(x)) + ((-1 - cot(x)^2)*sin(x))/cot(x)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \cot^{\sin{\left(x \right)} - 1}{\left(x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\left(\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \cot^{\sin{\left(x \right)} - 1}{\left(x \right)}}{\frac{d}{d x} \frac{1}{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{3} \cos^{3}{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 6 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 4 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \sin^{3}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \frac{\sin^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}} + \sin^{2}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \frac{\sin^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}}}{\left(\log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cot{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} - 2 \sin{\left(x \right)} \cot^{3}{\left(x \right)} - 2 \sin{\left(x \right)} \cot{\left(x \right)} + 2 \cos{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \cos{\left(x \right)}\right) \cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{3} \cos^{3}{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 6 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 4 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \sin^{3}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \frac{\sin^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}} + \sin^{2}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \frac{\sin^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}}}{\left(\log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cot{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} - 2 \sin{\left(x \right)} \cot^{3}{\left(x \right)} - 2 \sin{\left(x \right)} \cot{\left(x \right)} + 2 \cos{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \cos{\left(x \right)}\right) \cot{\left(x \right)}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \cot^{\sin{\left(x \right)} - 1}{\left(x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\left(\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \cot^{\sin{\left(x \right)} - 1}{\left(x \right)}}{\frac{d}{d x} \frac{1}{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{3} \cos^{3}{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 6 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 4 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \sin^{3}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \frac{\sin^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}} + \sin^{2}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \frac{\sin^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}}}{\left(\log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cot{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} - 2 \sin{\left(x \right)} \cot^{3}{\left(x \right)} - 2 \sin{\left(x \right)} \cot{\left(x \right)} + 2 \cos{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \cos{\left(x \right)}\right) \cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{3} \cos^{3}{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \log{\left(\cot{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 6 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \log{\left(\cot{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{4}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 4 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 2 \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \sin^{3}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - 3 \sin^{3}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} - \frac{\sin^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}} + \sin^{2}{\left(x \right)} \cot^{5}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot^{3}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cot{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)} + \frac{\sin^{2}{\left(x \right)} \cot^{\sin{\left(x \right)}}{\left(x \right)}}{\cot{\left(x \right)}}}{\left(\log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)} \cot{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)} - 2 \sin{\left(x \right)} \cot^{3}{\left(x \right)} - 2 \sin{\left(x \right)} \cot{\left(x \right)} + 2 \cos{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \cos{\left(x \right)}\right) \cot{\left(x \right)}}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^-}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right) = \infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right) = \infty$$
$$\lim_{x \to \infty}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)} \cos{\left(1 \right)} \tan{\left(1 \right)} + \sin{\left(1 \right)} + \sin{\left(1 \right)} \tan^{2}{\left(1 \right)}}{\tan{\left(1 \right)} \tan^{\sin{\left(1 \right)}}{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)} \cos{\left(1 \right)} \tan{\left(1 \right)} + \sin{\left(1 \right)} + \sin{\left(1 \right)} \tan^{2}{\left(1 \right)}}{\tan{\left(1 \right)} \tan^{\sin{\left(1 \right)}}{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right) = \infty$$
$$\lim_{x \to \infty}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)} \cos{\left(1 \right)} \tan{\left(1 \right)} + \sin{\left(1 \right)} + \sin{\left(1 \right)} \tan^{2}{\left(1 \right)}}{\tan{\left(1 \right)} \tan^{\sin{\left(1 \right)}}{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)} \cos{\left(1 \right)} \tan{\left(1 \right)} + \sin{\left(1 \right)} + \sin{\left(1 \right)} \tan^{2}{\left(1 \right)}}{\tan{\left(1 \right)} \tan^{\sin{\left(1 \right)}}{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ / / 2 \ \\
| sin(x) | \-1 - cot (x)/*sin(x)||
lim |cot (x)*|cos(x)*log(cot(x)) + ---------------------||
x->0+\ \ cot(x) //
$$\lim_{x \to 0^+}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
oo
$$\infty$$
= 7.88658225034602
/ / / 2 \ \\
| sin(x) | \-1 - cot (x)/*sin(x)||
lim |cot (x)*|cos(x)*log(cot(x)) + ---------------------||
x->0-\ \ cot(x) //
$$\lim_{x \to 0^-}\left(\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\sin{\left(x \right)}}{\left(x \right)}\right)$$
oo
$$\infty$$
= (7.85325563373746 + 3.07498909154237j)
= (7.85325563373746 + 3.07498909154237j)