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Límite de (1-cos(5*x))/x^2
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Expresiones idénticas
- x*(log(x)^ dos -log(dos *x))
- x multiplicar por ( logaritmo de (x) al cuadrado menos logaritmo de (2 multiplicar por x))
- x multiplicar por ( logaritmo de (x) en el grado dos menos logaritmo de (dos multiplicar por x))
- x*(log(x)2-log(2*x))
- x*logx2-log2*x
- x*(log(x)²-log(2*x))
- x*(log(x) en el grado 2-log(2*x))
- x(log(x)^2-log(2x))
- x(log(x)2-log(2x))
- xlogx2-log2x
- xlogx^2-log2x
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Expresiones semejantes
- x*(log(x)^2+log(2*x))
-
Expresiones con funciones
- Logaritmo log
- log(cos(3*x))/log(cos(2*x))
- log(sin(2*x))/log(sin(3*x))
- log(x)/x^(3/2)
- log(|x|)
- log(1-x)/(1+3*log(cos(pi*x/2)))
- Logaritmo log
- log(cos(3*x))/log(cos(2*x))
- log(sin(2*x))/log(sin(3*x))
- log(x)/x^(3/2)
- log(|x|)
- log(1-x)/(1+3*log(cos(pi*x/2)))
Límite de la función x*(log(x)^2-log(2*x))
Solución
Ha introducido
[src]
/ / 2 \\ lim \x*\log (x) - log(2*x)// x->0+
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right)$$
Limit(x*(log(x)^2 - log(2*x)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} x = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}^{2} - \log{\left(2 x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\log{\left(x \right)}^{2} - \log{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)^{2}}{- \frac{2 \log{\left(x \right)}}{x} + \frac{1}{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2 \log{\left(x \right)}}{x} + \frac{1}{x}}}{\frac{d}{d x} \frac{1}{\left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(- \frac{4 \log{\left(x \right)}}{x} + \frac{2}{x}\right) \left(\frac{4 \log{\left(x \right)}^{2}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} - \frac{4 \log{\left(x \right)}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} + \frac{1}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(- \frac{4 \log{\left(x \right)}}{x} + \frac{2}{x}\right) \left(\frac{4 \log{\left(x \right)}^{2}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} - \frac{4 \log{\left(x \right)}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} + \frac{1}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}}\right)}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} x = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}^{2} - \log{\left(2 x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\log{\left(x \right)}^{2} - \log{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)^{2}}{- \frac{2 \log{\left(x \right)}}{x} + \frac{1}{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2 \log{\left(x \right)}}{x} + \frac{1}{x}}}{\frac{d}{d x} \frac{1}{\left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(- \frac{4 \log{\left(x \right)}}{x} + \frac{2}{x}\right) \left(\frac{4 \log{\left(x \right)}^{2}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} - \frac{4 \log{\left(x \right)}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} + \frac{1}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(- \frac{4 \log{\left(x \right)}}{x} + \frac{2}{x}\right) \left(\frac{4 \log{\left(x \right)}^{2}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} - \frac{4 \log{\left(x \right)}}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}} + \frac{1}{- 2 \log{\left(x \right)}^{7} + 9 \log{\left(x \right)}^{6} - 15 \log{\left(x \right)}^{5} + 6 \log{\left(2 \right)} \log{\left(x \right)}^{5} - 21 \log{\left(2 \right)} \log{\left(x \right)}^{4} + 11 \log{\left(x \right)}^{4} - 3 \log{\left(x \right)}^{3} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 24 \log{\left(2 \right)} \log{\left(x \right)}^{3} - 9 \log{\left(2 \right)} \log{\left(x \right)}^{2} + 15 \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} - 9 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 \right)}^{3} \log{\left(x \right)} - 3 \log{\left(2 \right)}^{3}}\right)}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = 0$$
$$\lim_{x \to \infty}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = \infty$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = -\infty$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = 0$$
$$\lim_{x \to \infty}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = \infty$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right) = -\infty$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ / 2 \\ lim \x*\log (x) - log(2*x)// x->0+
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right)$$
0
$$0$$
= 0.0184658185740791
/ / 2 \\ lim \x*\log (x) - log(2*x)// x->0-
$$\lim_{x \to 0^-}\left(x \left(\log{\left(x \right)}^{2} - \log{\left(2 x \right)}\right)\right)$$
0
$$0$$
= (-0.0146692218311874 + 0.0129568416103419j)
= (-0.0146692218311874 + 0.0129568416103419j)