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sqrt(x)*exp(-x)

Limit of the function sqrt(x)*exp(-x)

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     /  ___  -x\
 lim \\/ x *e  /
x->oo           
limx(xex)\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right)
Limit(sqrt(x)*exp(-x), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
limxx=\lim_{x \to \infty} \sqrt{x} = \infty
and limit for the denominator is
limxex=\lim_{x \to \infty} e^{x} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
limx(xex)\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right)
=
limx(ddxxddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \sqrt{x}}{\frac{d}{d x} e^{x}}\right)
=
limx(ex2x)\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{x}}\right)
=
limx(ex2x)\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{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
02468-8-6-4-2-10100.00.5
Rapid solution [src]
0
00
Other limits x→0, -oo, +oo, 1
limx(xex)=0\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right) = 0
limx0(xex)=0\lim_{x \to 0^-}\left(\sqrt{x} e^{- x}\right) = 0
More at x→0 from the left
limx0+(xex)=0\lim_{x \to 0^+}\left(\sqrt{x} e^{- x}\right) = 0
More at x→0 from the right
limx1(xex)=e1\lim_{x \to 1^-}\left(\sqrt{x} e^{- x}\right) = e^{-1}
More at x→1 from the left
limx1+(xex)=e1\lim_{x \to 1^+}\left(\sqrt{x} e^{- x}\right) = e^{-1}
More at x→1 from the right
limx(xex)=i\lim_{x \to -\infty}\left(\sqrt{x} e^{- x}\right) = \infty i
More at x→-oo
The graph
Limit of the function sqrt(x)*exp(-x)