Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (2*x/(1+2*x))^x
-
Límite de (5+x)/(-6+3*x)
-
Límite de (1-sqrt(1-x^2))/x^2
-
Límite de (-2+x^3-3*x)/(-2+x)
-
Expresiones idénticas
- cos(cuatro *x)*cot(cinco *x)*tan(tres *x)/(cos(siete *x)*cot(siete *x)*sin(seis *x))
- coseno de (4 multiplicar por x) multiplicar por cotangente de (5 multiplicar por x) multiplicar por tangente de (3 multiplicar por x) dividir por ( coseno de (7 multiplicar por x) multiplicar por cotangente de (7 multiplicar por x) multiplicar por seno de (6 multiplicar por x))
- coseno de (cuatro multiplicar por x) multiplicar por cotangente de (cinco multiplicar por x) multiplicar por tangente de (tres multiplicar por x) dividir por ( coseno de (siete multiplicar por x) multiplicar por cotangente de (siete multiplicar por x) multiplicar por seno de (seis multiplicar por x))
- cos(4x)cot(5x)tan(3x)/(cos(7x)cot(7x)sin(6x))
- cos4xcot5xtan3x/cos7xcot7xsin6x
- cos(4*x)*cot(5*x)*tan(3*x) dividir por (cos(7*x)*cot(7*x)*sin(6*x))
-
Expresiones con funciones
- Coseno cos
- cos(sqrt(x))/x
- cos(x)/(1-sin(x))
- cos(x)/(1+x)
- cos(sqrt(x))
- cos(pi/(6*x))
- Cotangente cot
- cot(7*x)^tan(x)
- cot(t)*log(t*cot(t))
- cot(9*x)^(1/log(x))
- cot(x)/log(10)
- cot(x)^cot(x)
- Tangente tan
- tan(x)/(3*x)
- tan(8*x)/x
- tan(2*x)/(5*x)
- tan(-3+x)/(-9+x^2)
- tan(x)^2-cos(x)
- Coseno cos
- cos(sqrt(x))/x
- cos(x)/(1-sin(x))
- cos(x)/(1+x)
- cos(sqrt(x))
- cos(pi/(6*x))
- Cotangente cot
- cot(7*x)^tan(x)
- cot(t)*log(t*cot(t))
- cot(9*x)^(1/log(x))
- cot(x)/log(10)
- cot(x)^cot(x)
- Seno sin
- sin(1)/x
- sin(7*pi*x)/sin(8*pi*x)
- sin(7*x)/(5*x)
- sin(x)/(-sin(7*x)+sin(6*x))
- sin(-1+x)/(-1+x^2)
Límite de la función cos(4*x)*cot(5*x)*tan(3*x)/(cos(7*x)*cot(7*x)*sin(6*x))
Solución
Ha introducido
[src]
/cos(4*x)*cot(5*x)*tan(3*x)\ lim |--------------------------| x->0+\cos(7*x)*cot(7*x)*sin(6*x)/
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
Limit(((cos(4*x)*cot(5*x))*tan(3*x))/(((cos(7*x)*cot(7*x))*sin(6*x))), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(7 x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}{\sin{\left(6 x \right)} \cos{\left(7 x \right)} \cot{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(7 x \right)}}}{\frac{d}{d x} \frac{\sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cot^{2}{\left(7 x \right)} + 7}{\left(\frac{4 \sin{\left(4 x \right)} \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos^{2}{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{7 \sin{\left(6 x \right)} \sin{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \cot{\left(5 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot^{2}{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan^{2}{\left(3 x \right)} \cot{\left(5 x \right)}} + \frac{6 \cos{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}\right) \cot^{2}{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cot^{2}{\left(7 x \right)} + 7}{\left(\frac{4 \sin{\left(4 x \right)} \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos^{2}{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{7 \sin{\left(6 x \right)} \sin{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \cot{\left(5 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot^{2}{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan^{2}{\left(3 x \right)} \cot{\left(5 x \right)}} + \frac{6 \cos{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}\right) \cot^{2}{\left(7 x \right)}}\right)$$
=
$$\frac{7}{10}$$
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
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(7 x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}{\sin{\left(6 x \right)} \cos{\left(7 x \right)} \cot{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(7 x \right)}}}{\frac{d}{d x} \frac{\sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cot^{2}{\left(7 x \right)} + 7}{\left(\frac{4 \sin{\left(4 x \right)} \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos^{2}{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{7 \sin{\left(6 x \right)} \sin{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \cot{\left(5 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot^{2}{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan^{2}{\left(3 x \right)} \cot{\left(5 x \right)}} + \frac{6 \cos{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}\right) \cot^{2}{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cot^{2}{\left(7 x \right)} + 7}{\left(\frac{4 \sin{\left(4 x \right)} \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos^{2}{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{7 \sin{\left(6 x \right)} \sin{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \cot{\left(5 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)}} + \frac{5 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot^{2}{\left(5 x \right)}} - \frac{3 \sin{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan^{2}{\left(3 x \right)} \cot{\left(5 x \right)}} + \frac{6 \cos{\left(6 x \right)} \cos{\left(7 x \right)}}{\cos{\left(4 x \right)} \tan{\left(3 x \right)} \cot{\left(5 x \right)}}\right) \cot^{2}{\left(7 x \right)}}\right)$$
=
$$\frac{7}{10}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/cos(4*x)*cot(5*x)*tan(3*x)\ lim |--------------------------| x->0+\cos(7*x)*cot(7*x)*sin(6*x)/
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
7/10
$$\frac{7}{10}$$
= 0.7
/cos(4*x)*cot(5*x)*tan(3*x)\ lim |--------------------------| x->0-\cos(7*x)*cot(7*x)*sin(6*x)/
$$\lim_{x \to 0^-}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
7/10
$$\frac{7}{10}$$
= 0.7
= 0.7
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right) = \frac{7}{10}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right) = \frac{7}{10}$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right) = \frac{\cos{\left(4 \right)} \tan{\left(3 \right)} \tan{\left(7 \right)}}{\sin{\left(6 \right)} \cos{\left(7 \right)} \tan{\left(5 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right) = \frac{\cos{\left(4 \right)} \tan{\left(3 \right)} \tan{\left(7 \right)}}{\sin{\left(6 \right)} \cos{\left(7 \right)} \tan{\left(5 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right) = \frac{7}{10}$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right) = \frac{\cos{\left(4 \right)} \tan{\left(3 \right)} \tan{\left(7 \right)}}{\sin{\left(6 \right)} \cos{\left(7 \right)} \tan{\left(5 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right) = \frac{\cos{\left(4 \right)} \tan{\left(3 \right)} \tan{\left(7 \right)}}{\sin{\left(6 \right)} \cos{\left(7 \right)} \tan{\left(5 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(5 x \right)} \tan{\left(3 x \right)}}{\cos{\left(7 x \right)} \cot{\left(7 x \right)} \sin{\left(6 x \right)}}\right)$$
Más detalles con x→-oo
Gráfico