Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-1+e^(2*x))/(3*x)
-
Limit of (-1+3*x)/(5+x^2+7*x)
-
Limit of 1+13*x/5
-
Limit of (7+x+x^2)/(-1+e^x)
- Graphing y =:
- (-x+atan(x))/x^3
-
Identical expressions
- (-x+atan(x))/x^ three
- ( minus x plus arc tangent of gent of (x)) divide by x cubed
- ( minus x plus arc tangent of gent of (x)) divide by x to the power of three
- (-x+atan(x))/x3
- -x+atanx/x3
- (-x+atan(x))/x³
- (-x+atan(x))/x to the power of 3
- -x+atanx/x^3
- (-x+atan(x)) divide by x^3
-
Similar expressions
- (x+atan(x))/x^3
- (-x-atan(x))/x^3
- (-x+arctan(x))/x^3
- (-x+arctanx)/x^3
Limit of the function (-x+atan(x))/x^3
The solution
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[src]
/-x + atan(x)\
lim |------------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
Limit((-x + atan(x))/(x^3), 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
$$\lim_{x \to 0^+} x^{3} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \operatorname{atan}{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{-1 + \frac{1}{x^{2} + 1}}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(-1 + \frac{1}{x^{2} + 1}\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{3 \left(x^{2} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{3}$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{3}$$
=
$$\lim_{x \to 0^+} - \frac{1}{3}$$
=
$$- \frac{1}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 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
$$\lim_{x \to 0^+} x^{3} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \operatorname{atan}{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{-1 + \frac{1}{x^{2} + 1}}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(-1 + \frac{1}{x^{2} + 1}\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{3 \left(x^{2} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{3}$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{3}$$
=
$$\lim_{x \to 0^+} - \frac{1}{3}$$
=
$$- \frac{1}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
One‐sided limits
[src]
/-x + atan(x)\
lim |------------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
-1/3
$$- \frac{1}{3}$$
= -0.333333333333333
/-x + atan(x)\
lim |------------|
x->0-| 3 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
-1/3
$$- \frac{1}{3}$$
= -0.333333333333333
= -0.333333333333333
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = - \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = - \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = -1 + \frac{\pi}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = -1 + \frac{\pi}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = - \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = -1 + \frac{\pi}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = -1 + \frac{\pi}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = 0$$
More at x→-oo
The graph