Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (1+x^2+9*x)/(-5+2*x+7*x^2)
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Limit of (-tan(2*x)+sin(2*x))/x^3
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Limit of 3/n^4
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Limit of (1-cos(x)^2)/(x^2*sin(x)^2)
- Derivative of:
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x^3-3*x
- Graphing y =:
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x^3-3*x
- Integral of d{x}:
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x^3-3*x
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Identical expressions
- x^ three - three *x
- x cubed minus 3 multiply by x
- x to the power of three minus three multiply by x
- x3-3*x
- x³-3*x
- x to the power of 3-3*x
- x^3-3x
- x3-3x
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Similar expressions
- (4*x+7*x^3)/(2+x^3-3*x)
- (4+x^3-3*x^2)/(1+x^5)
- (-1+(1+x)^3-3*x)/(x+x^3)
- x^3+3*x
Limit of the function x^3-3*x
The solution
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/ 3 \ lim \x - 3*x/ x->oo
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right)$$
Limit(x^3 - 3*x, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{3}{x^{2}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{3}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{1 - 3 u^{2}}{u^{3}}\right)$$
=
$$\frac{1 - 3 \cdot 0^{2}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{3}{x^{2}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{3}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{1 - 3 u^{2}}{u^{3}}\right)$$
=
$$\frac{1 - 3 \cdot 0^{2}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right) = \infty$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x^{3} - 3 x\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{3} - 3 x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} - 3 x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} - 3 x\right) = -2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} - 3 x\right) = -2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} - 3 x\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{3} - 3 x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} - 3 x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} - 3 x\right) = -2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} - 3 x\right) = -2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} - 3 x\right) = -\infty$$
More at x→-oo
The graph