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

You have entered [src]

     /  ___  -x\
 lim \\/ x *e  /
x->oo

\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

\lim_{x \to \infty} \sqrt{x} = \infty

and limit for the denominator is

\lim_{x \to \infty} e^{x} = \infty

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \sqrt{x}}{\frac{d}{d x} e^{x}}\right)

\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{x}}\right)

\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{x}}\right)

0

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

Plot the graph

Rapid solution [src]

0

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right) = 0

\lim_{x \to 0^-}\left(\sqrt{x} e^{- x}\right) = 0

\lim_{x \to 0^+}\left(\sqrt{x} e^{- x}\right) = 0

\lim_{x \to 1^-}\left(\sqrt{x} e^{- x}\right) = e^{-1}

\lim_{x \to 1^+}\left(\sqrt{x} e^{- x}\right) = e^{-1}

\lim_{x \to -\infty}\left(\sqrt{x} e^{- x}\right) = \infty i

The graph