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

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

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You have entered [src] 
     /     2\
     |   -x |
 lim \x*e   /
x->oo
lim⁡x→∞(xe−x2)\lim_{x \to \infty}\left(x e^{- x^{2}}\right) 
Limit(x*exp(-x^2), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞x=∞\lim_{x \to \infty} x = \infty
and limit for the denominator is
lim⁡x→∞ex2=∞\lim_{x \to \infty} e^{x^{2}} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(xe−x2)\lim_{x \to \infty}\left(x e^{- x^{2}}\right)
=
lim⁡x→∞(ddxxddxex2)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{x^{2}}}\right)
=
lim⁡x→∞(e−x22x)\lim_{x \to \infty}\left(\frac{e^{- x^{2}}}{2 x}\right)
=
lim⁡x→∞(e−x22x)\lim_{x \to \infty}\left(\frac{e^{- x^{2}}}{2 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) 1 time(s)
The graph
02468-8-6-4-2-10101.0-1.0
Plot the graph
Rapid solution [src] 
0
00
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(xe−x2)=0\lim_{x \to \infty}\left(x e^{- x^{2}}\right) = 0
lim⁡x→0−(xe−x2)=0\lim_{x \to 0^-}\left(x e^{- x^{2}}\right) = 0
More at x→0 from the left
lim⁡x→0+(xe−x2)=0\lim_{x \to 0^+}\left(x e^{- x^{2}}\right) = 0
More at x→0 from the right
lim⁡x→1−(xe−x2)=e−1\lim_{x \to 1^-}\left(x e^{- x^{2}}\right) = e^{-1}
More at x→1 from the left
lim⁡x→1+(xe−x2)=e−1\lim_{x \to 1^+}\left(x e^{- x^{2}}\right) = e^{-1}
More at x→1 from the right
lim⁡x→−∞(xe−x2)=0\lim_{x \to -\infty}\left(x e^{- x^{2}}\right) = 0
More at x→-oo
The graph
Limit of the function x*exp(-x^2)
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