Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of x^2*(1/3-cos(8*x)/3)
-
Limit of (-2+x^2-x)/(-2+x+3*x^2)
-
Limit of (-1+sin(x))/cos(x)
-
Limit of (-2*sin(x)+sin(2*x))/(x*log(cos(5*x)))
-
Identical expressions
- sin(x)^ four /x^ four
- sinus of (x) to the power of 4 divide by x to the power of 4
- sinus of (x) to the power of four divide by x to the power of four
- sin(x)4/x4
- sinx4/x4
- sin(x)⁴/x⁴
- sinx^4/x^4
- sin(x)^4 divide by x^4
-
Similar expressions
- sinx^4/x^4
-
What you mean?
- sin(x)^4/(x^4)
- sin(x)^((4/x)^4)
- sin(x)^((4/x)^4)
You entered:
sin(x)^4/x^4
What you mean?
Limit of the function sin(x)^4/x^4
The solution
You have entered
[src]
/ 4 \
|sin (x)|
lim |-------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)$$
Limit(sin(x)^4/(x^4), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin^{4}{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{4} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin^{4}{\left(x \right)}}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{3}{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 4 \sin^{3}{\left(x \right)}}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 12 \sin^{2}{\left(x \right)}}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 24 \sin{\left(x \right)}}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 4 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin^{4}{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{4} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin^{4}{\left(x \right)}}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{3}{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 4 \sin^{3}{\left(x \right)}}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 12 \sin^{2}{\left(x \right)}}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 24 \sin{\left(x \right)}}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 4 time(s)
The graph
One‐sided limits
[src]
/ 4 \
|sin (x)|
lim |-------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)$$
1
$$1$$
= 1
/ 4 \
|sin (x)|
lim |-------|
x->0-| 4 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)$$
1
$$1$$
= 1
= 1
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = \sin^{4}{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = \sin^{4}{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = \sin^{4}{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = \sin^{4}{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→-oo
The graph