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

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

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You have entered [src] 
     /   -x\
 lim \x*E  /
x->oo
lim⁡x→∞(e−xx)\lim_{x \to \infty}\left(e^{- x} x\right) 
Limit(x*E^(-x), 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→∞ex=∞\lim_{x \to \infty} e^{x} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(e−xx)\lim_{x \to \infty}\left(e^{- x} x\right)
=
Let's transform the function under the limit a few
lim⁡x→∞(xe−x)\lim_{x \to \infty}\left(x e^{- x}\right)
=
lim⁡x→∞(ddxxddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{x}}\right)
=
lim⁡x→∞e−x\lim_{x \to \infty} e^{- x}
=
lim⁡x→∞e−x\lim_{x \to \infty} e^{- x}
=
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
02468-8-6-4-2-1010-200000200000
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(e−xx)=0\lim_{x \to \infty}\left(e^{- x} x\right) = 0
lim⁡x→0−(e−xx)=0\lim_{x \to 0^-}\left(e^{- x} x\right) = 0
More at x→0 from the left
lim⁡x→0+(e−xx)=0\lim_{x \to 0^+}\left(e^{- x} x\right) = 0
More at x→0 from the right
lim⁡x→1−(e−xx)=e−1\lim_{x \to 1^-}\left(e^{- x} x\right) = e^{-1}
More at x→1 from the left
lim⁡x→1+(e−xx)=e−1\lim_{x \to 1^+}\left(e^{- x} x\right) = e^{-1}
More at x→1 from the right
lim⁡x→−∞(e−xx)=−∞\lim_{x \to -\infty}\left(e^{- x} x\right) = -\infty
More at x→-oo
Rapid solution [src] 
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00
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The graph
Limit of the function x*e^(-x)
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