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e^(-x)*x^3
Limit of the function/ e^(-x)*x^3

Limit of the function e^(-x)*x^3

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     / -x  3\
 lim \E  *x /
x->oo
lim⁡x→∞(e−xx3)\lim_{x \to \infty}\left(e^{- x} x^{3}\right) 
Limit(E^(-x)*x^3, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞x3=∞\lim_{x \to \infty} x^{3} = \infty
and limit for the denominator is
lim⁡x→∞ex=∞\lim_{x \to \infty} e^{x} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(e−xx3)\lim_{x \to \infty}\left(e^{- x} x^{3}\right)
=
Let's transform the function under the limit a few
lim⁡x→∞(x3e−x)\lim_{x \to \infty}\left(x^{3} e^{- x}\right)
=
lim⁡x→∞(ddxx3ddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{3}}{\frac{d}{d x} e^{x}}\right)
=
lim⁡x→∞(3x2e−x)\lim_{x \to \infty}\left(3 x^{2} e^{- x}\right)
=
lim⁡x→∞(ddx3x2ddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 3 x^{2}}{\frac{d}{d x} e^{x}}\right)
=
lim⁡x→∞(6xe−x)\lim_{x \to \infty}\left(6 x e^{- x}\right)
=
lim⁡x→∞(ddx6xddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 6 x}{\frac{d}{d x} e^{x}}\right)
=
lim⁡x→∞(6e−x)\lim_{x \to \infty}\left(6 e^{- x}\right)
=
lim⁡x→∞(6e−x)\lim_{x \to \infty}\left(6 e^{- x}\right)
=
00
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
02468-8-6-4-2-1010-2000000020000000
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(e−xx3)=0\lim_{x \to \infty}\left(e^{- x} x^{3}\right) = 0
lim⁡x→0−(e−xx3)=0\lim_{x \to 0^-}\left(e^{- x} x^{3}\right) = 0
More at x→0 from the left
lim⁡x→0+(e−xx3)=0\lim_{x \to 0^+}\left(e^{- x} x^{3}\right) = 0
More at x→0 from the right
lim⁡x→1−(e−xx3)=e−1\lim_{x \to 1^-}\left(e^{- x} x^{3}\right) = e^{-1}
More at x→1 from the left
lim⁡x→1+(e−xx3)=e−1\lim_{x \to 1^+}\left(e^{- x} x^{3}\right) = e^{-1}
More at x→1 from the right
lim⁡x→−∞(e−xx3)=−∞\lim_{x \to -\infty}\left(e^{- x} x^{3}\right) = -\infty
More at x→-oo
Rapid solution [src] 
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The graph
Limit of the function e^(-x)*x^3
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