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x^3-9*x
Limit of the function/ x^3-9*x

Limit of the function x^3-9*x

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     / 3      \
 lim \x  - 9*x/
x->oo
lim⁡x→∞(x3−9x)\lim_{x \to \infty}\left(x^{3} - 9 x\right) 
Limit(x^3 - 9*x, x, oo, dir='-')
Detail solution
Let's take the limit
lim⁡x→∞(x3−9x)\lim_{x \to \infty}\left(x^{3} - 9 x\right)
Let's divide numerator and denominator by x^3:
lim⁡x→∞(x3−9x)\lim_{x \to \infty}\left(x^{3} - 9 x\right) =
lim⁡x→∞(1−9x21x3)\lim_{x \to \infty}\left(\frac{1 - \frac{9}{x^{2}}}{\frac{1}{x^{3}}}\right)
Do Replacement
u=1xu = \frac{1}{x}
then
lim⁡x→∞(1−9x21x3)=lim⁡u→0+(1−9u2u3)\lim_{x \to \infty}\left(\frac{1 - \frac{9}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{1 - 9 u^{2}}{u^{3}}\right)
=
1−9⋅020=∞\frac{1 - 9 \cdot 0^{2}}{0} = \infty

The final answer:
lim⁡x→∞(x3−9x)=∞\lim_{x \to \infty}\left(x^{3} - 9 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
02468-8-6-4-2-1010-20002000
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Rapid solution [src] 
oo
∞\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(x3−9x)=∞\lim_{x \to \infty}\left(x^{3} - 9 x\right) = \infty
lim⁡x→0−(x3−9x)=0\lim_{x \to 0^-}\left(x^{3} - 9 x\right) = 0
More at x→0 from the left
lim⁡x→0+(x3−9x)=0\lim_{x \to 0^+}\left(x^{3} - 9 x\right) = 0
More at x→0 from the right
lim⁡x→1−(x3−9x)=−8\lim_{x \to 1^-}\left(x^{3} - 9 x\right) = -8
More at x→1 from the left
lim⁡x→1+(x3−9x)=−8\lim_{x \to 1^+}\left(x^{3} - 9 x\right) = -8
More at x→1 from the right
lim⁡x→−∞(x3−9x)=−∞\lim_{x \to -\infty}\left(x^{3} - 9 x\right) = -\infty
More at x→-oo
The graph
Limit of the function x^3-9*x
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