Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (-64+4^x)/(-3+x)
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Limit of (3-x^2+5*x)/(4*x^7+81*x)
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Limit of (-2+2*x^3+7*x)/(-4-x+3*x^3)
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Limit of (1-sin(x))/cos(x)
- Graphing y =:
- -x^3+3*x
- Derivative of:
- -x^3+3*x
- Integral of d{x}:
- -x^3+3*x
-
Identical expressions
- -x^ three + three *x
- minus x cubed plus 3 multiply by x
- minus x to the power of three plus three multiply by x
- -x3+3*x
- -x³+3*x
- -x to the power of 3+3*x
- -x^3+3x
- -x3+3x
-
Similar expressions
- -x^3-3*x
- 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)
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
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{3}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{3 u^{2} - 1}{u^{3}}\right)$$
=
$$\frac{-1 + 3 \cdot 0^{2}}{0} = \infty$$
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
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{3}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{3 u^{2} - 1}{u^{3}}\right)$$
=
$$\frac{-1 + 3 \cdot 0^{2}}{0} = \infty$$
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
$$\lim_{x \to -\infty}\left(- x^{3} + 3 x\right) = \infty$$
$$\lim_{x \to \infty}\left(- x^{3} + 3 x\right) = -\infty$$
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
$$\lim_{x \to 0^-}\left(- x^{3} + 3 x\right) = 0$$
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
The graph