Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of ((3+x)^2+(3-x)^2)/((3-x)^2-(3+x)^2)
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Limit of (-1+x)/(-2+sqrt(3+x))
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Limit of (-x+tan(x))/x^3
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Limit of (-1+sqrt(x))/(-3+x)
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Identical expressions
- (-x+tan(x))/x^ three
- ( minus x plus tangent of (x)) divide by x cubed
- ( minus x plus tangent of (x)) divide by x to the power of three
- (-x+tan(x))/x3
- -x+tanx/x3
- (-x+tan(x))/x³
- (-x+tan(x))/x to the power of 3
- -x+tanx/x^3
- (-x+tan(x)) divide by x^3
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Similar expressions
- (x+tan(x))/x^3
- (-x-tan(x))/x^3
Limit of the function (-x+tan(x))/x^3
The solution
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[src]
/-x + tan(x)\
lim |-----------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
Limit((-x + tan(x))/x^3, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- x + \tan{\left(x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \tan{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)}}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan^{2}{\left(x \right)}}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{3 x}\right)$$
=
$$\frac{1}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- x + \tan{\left(x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \tan{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)}}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan^{2}{\left(x \right)}}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{3 x}\right)$$
=
$$\frac{1}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
One‐sided limits
[src]
/-x + tan(x)\
lim |-----------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
/-x + tan(x)\
lim |-----------|
x->0-| 3 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
= 0.333333333333333
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = -1 + \tan{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = -1 + \tan{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = -1 + \tan{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = -1 + \tan{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→-oo
The graph