Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of -2+x^3+6*x
-
Limit of ((-2+x)/x)^(2*x)
-
Limit of (-2+x)*log(-2+x)
-
Limit of -6+x^2+5*x
- Graphing y =:
- x^3-12*x
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Identical expressions
- x^ three - twelve *x
- x cubed minus 12 multiply by x
- x to the power of three minus twelve multiply by x
- x3-12*x
- x³-12*x
- x to the power of 3-12*x
- x^3-12x
- x3-12x
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Similar expressions
- x^3+12*x
Limit of the function x^3-12*x
The solution
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/ 3 \ lim \x - 12*x/ x->oo
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right)$$
Limit(x^3 - 12*x, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{12}{x^{2}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{12}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{1 - 12 u^{2}}{u^{3}}\right)$$
=
$$\frac{1 - 12 \cdot 0^{2}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{12}{x^{2}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{12}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{1 - 12 u^{2}}{u^{3}}\right)$$
=
$$\frac{1 - 12 \cdot 0^{2}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right) = \infty$$
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
$$\lim_{x \to \infty}\left(x^{3} - 12 x\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{3} - 12 x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} - 12 x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} - 12 x\right) = -11$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} - 12 x\right) = -11$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} - 12 x\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{3} - 12 x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} - 12 x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} - 12 x\right) = -11$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} - 12 x\right) = -11$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} - 12 x\right) = -\infty$$
More at x→-oo
The graph