Sr Examen
Otras calculadoras:
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- Límite de la función:
-
Límite de x^x
-
Límite de (-4+x^2)/(-2+x)
-
Límite de (-sin(x)+tan(x))/x^3
-
Límite de x*sin(1/x)
-
Expresiones idénticas
- cot(x)*log(x+e^x)
- cotangente de (x) multiplicar por logaritmo de (x más e en el grado x)
- cot(x)*log(x+ex)
- cotx*logx+ex
- cot(x)log(x+e^x)
- cot(x)log(x+ex)
- cotxlogx+ex
- cotxlogx+e^x
-
Expresiones semejantes
- cot(x)*log(x-e^x)
-
Expresiones con funciones
- Cotangente cot
- cot(x)^x
- cot(2*x)/cot(x)
- cot(x)^sin(x)
- cot(x)^2
- cot(2*x)*tan(x)
- Logaritmo log
- log(x)
- log(cot(x))*sin(x)
- log(1+n)
- log(1+x)
- log(x)/log(cot(x))
Límite de la función cot(x)*log(x+e^x)
Solución
Ha introducido
[src]
/ / x\\ lim \cot(x)*log\x + E // x->0+
$$\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
Limit(cot(x)*log(x + E^x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \log{\left(x + e^{x} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\log{\left(x + e^{x} \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(x + e^{x} \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(e^{x} + 1\right) \cot^{2}{\left(x \right)}}{\left(x + e^{x}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$2$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \log{\left(x + e^{x} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\log{\left(x + e^{x} \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(x + e^{x} \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(e^{x} + 1\right) \cot^{2}{\left(x \right)}}{\left(x + e^{x}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$2$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = 2$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(1 + e \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(1 + e \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(1 + e \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(1 + e \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ / x\\ lim \cot(x)*log\x + E // x->0+
$$\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
2
$$2$$
= 2.0
/ / x\\ lim \cot(x)*log\x + E // x->0-
$$\lim_{x \to 0^-}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)$$
2
$$2$$
= 2.0
= 2.0
Gráfico