What is the limit as x approaches infinity of sqrt(x)*e^{-x/2} ?

The limit as x approaches infinity of sqrt(x)*e^{-x/2} is 0

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limit as x approaches infinity of sqrt(x)*e^{-x/2}

x \to \infty lim (x \cdot e^{- \frac{x}{2}})

0

Solution steps

x \to \infty lim (x e^{- \frac{x}{2}})

Simplify

e^{- \frac{x}{2}} : \frac{1}{e ^{\frac{x}{2}}}

= x \to \infty lim (x \frac{1}{e ^{\frac{x}{2}}})

Simplify

x \frac{1}{e ^{\frac{x}{2}}} : \frac{x}{e ^{x}}

= x \to \infty lim (\frac{x}{e ^{x}})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

= x \to \infty lim (\frac{x}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{1}{e ^{x}})

= x \to \infty lim (\frac{1}{e ^{x}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( e ^{x} )}

x \to \infty lim (1) = 1

x \to \infty lim (e^{x}) = \infty

= \frac{1}{\infty}

Simplify

\frac{1}{\infty} : 0

= 0

y = 4 - x^{2}, y = x^{2} - 4

\frac{d}{d x} (C x e^{4 x})

\int \frac{x}{x + 11} d x

x y^{^{'}} + y = e x

n = 1 \sum \infty \frac{3 n}{n + 1}

What is the limit as x approaches infinity of sqrt(x)*e^{-x/2} ?
The limit as x approaches infinity of sqrt(x)*e^{-x/2} is 0