Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of -2+x^3+6*x
-
Limit of ((-2+x)/x)^(2*x)
-
Limit of -6+x^2+5*x
-
Limit of (2+x^2-3*x)/(4+x^2-5*x)
- Graphing y =:
- (x-atan(x))/x^4
-
Identical expressions
- (x-atan(x))/x^ four
- (x minus arc tangent of gent of (x)) divide by x to the power of 4
- (x minus arc tangent of gent of (x)) divide by x to the power of four
- (x-atan(x))/x4
- x-atanx/x4
- (x-atan(x))/x⁴
- x-atanx/x^4
- (x-atan(x)) divide by x^4
-
Similar expressions
- (x+atan(x))/x^4
- (x-arctan(x))/x^4
- (x-arctanx)/x^4
Limit of the function (x-atan(x))/x^4
The solution
You have entered
[src]
/x - atan(x)\
lim |-----------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right)$$
Limit((x - atan(x))/x^4, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(x - \operatorname{atan}{\left(x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x - \operatorname{atan}{\left(x \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1 - \frac{1}{x^{2} + 1}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \frac{1}{x^{2} + 1}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{6 x \left(x^{2} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{6 x}\right)$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(x - \operatorname{atan}{\left(x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x - \operatorname{atan}{\left(x \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1 - \frac{1}{x^{2} + 1}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \frac{1}{x^{2} + 1}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{6 x \left(x^{2} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{6 x}\right)$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 1 - \frac{\pi}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 1 - \frac{\pi}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 1 - \frac{\pi}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 1 - \frac{\pi}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/x - atan(x)\
lim |-----------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right)$$
oo
$$\infty$$
= 50.3320088715133
/x - atan(x)\
lim |-----------|
x->0-| 4 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right)$$
-oo
$$-\infty$$
= -50.3320088715133
= -50.3320088715133
The graph