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- Límite de la función:
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Límite de (-3+x)/(-9+x^2)
-
Límite de x^sin(x)
- Límite de (-9+x^2)/(6+x^2-5*x)
-
Límite de (1+x^2-2*x)/(x^3-x)
-
Expresiones idénticas
- (- uno + dos ^x)*cot(x)
- ( menos 1 más 2 en el grado x) multiplicar por cotangente de (x)
- ( menos uno más dos en el grado x) multiplicar por cotangente de (x)
- (-1+2x)*cot(x)
- -1+2x*cotx
- (-1+2^x)cot(x)
- (-1+2x)cot(x)
- -1+2xcotx
- -1+2^xcotx
-
Expresiones semejantes
- (-1-2^x)*cot(x)
- (1+2^x)*cot(x)
-
Expresiones con funciones
- Cotangente cot
- cot(2*x)
- cot(pi*x/2)/log(-2+x)
- cot(3*x)/cot(6*x)
- cot(x+pi/4)*tan(2*x)
- cot(x)*log(x)*sin(x)
Límite de la función (-1+2^x)*cot(x)
Solución
Ha introducido
[src]
// x\ \ lim \\-1 + 2 /*cot(x)/ x->0+
$$\lim_{x \to 0^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
Limit((-1 + 2^x)*cot(x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(2^{x} - 1\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2^{x} - 1\right)}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2^{x} \log{\left(2 \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(2 \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{\log{\left(2 \right)} \cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}}\right)}\right)$$
=
$$\log{\left(2 \right)}$$
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
$$\lim_{x \to 0^+}\left(2^{x} - 1\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2^{x} - 1\right)}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2^{x} \log{\left(2 \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(2 \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{\log{\left(2 \right)} \cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}} - \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \log{\left(2 \right)} \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \log{\left(2 \right)} \cot^{3}{\left(x \right)}}{\left(- 2 \log{\left(2 \right)} \cot^{6}{\left(x \right)} - 2 \log{\left(2 \right)} \cot^{4}{\left(x \right)}\right)^{2}}\right)}\right)$$
=
$$\log{\left(2 \right)}$$
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
$$\lim_{x \to 0^-}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right) = \log{\left(2 \right)}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right) = \log{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right) = \frac{1}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right) = \frac{1}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right) = \log{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right) = \frac{1}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right) = \frac{1}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
// x\ \ lim \\-1 + 2 /*cot(x)/ x->0+
$$\lim_{x \to 0^+}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
log(2)
$$\log{\left(2 \right)}$$
= 0.693147180559945
// x\ \ lim \\-1 + 2 /*cot(x)/ x->0-
$$\lim_{x \to 0^-}\left(\left(2^{x} - 1\right) \cot{\left(x \right)}\right)$$
log(2)
$$\log{\left(2 \right)}$$
= 0.693147180559945
= 0.693147180559945
Gráfico