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x^2*exp(-x)
Limit of the function/ x^2*exp(-x)

Limit of the function x^2*exp(-x)

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     / 2  -x\
 lim \x *e  /
x->oo
lim⁡x→∞(x2e−x)\lim_{x \to \infty}\left(x^{2} e^{- x}\right) 
Limit(x^2*exp(-x), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞x2=∞\lim_{x \to \infty} x^{2} = \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→∞(x2e−x)\lim_{x \to \infty}\left(x^{2} e^{- x}\right)
=
lim⁡x→∞(ddxx2ddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{2}}{\frac{d}{d x} e^{x}}\right)
=
lim⁡x→∞(2xe−x)\lim_{x \to \infty}\left(2 x e^{- x}\right)
=
lim⁡x→∞(ddx2xddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 x}{\frac{d}{d x} e^{x}}\right)
=
lim⁡x→∞(2e−x)\lim_{x \to \infty}\left(2 e^{- x}\right)
=
lim⁡x→∞(2e−x)\lim_{x \to \infty}\left(2 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) 2 time(s)
The graph
02468-8-6-4-2-101002000000
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(x2e−x)=0\lim_{x \to \infty}\left(x^{2} e^{- x}\right) = 0
lim⁡x→0−(x2e−x)=0\lim_{x \to 0^-}\left(x^{2} e^{- x}\right) = 0
More at x→0 from the left
lim⁡x→0+(x2e−x)=0\lim_{x \to 0^+}\left(x^{2} e^{- x}\right) = 0
More at x→0 from the right
lim⁡x→1−(x2e−x)=e−1\lim_{x \to 1^-}\left(x^{2} e^{- x}\right) = e^{-1}
More at x→1 from the left
lim⁡x→1+(x2e−x)=e−1\lim_{x \to 1^+}\left(x^{2} e^{- x}\right) = e^{-1}
More at x→1 from the right
lim⁡x→−∞(x2e−x)=∞\lim_{x \to -\infty}\left(x^{2} e^{- x}\right) = \infty
More at x→-oo
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00
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Limit of the function x^2*exp(-x)
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