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

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

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

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

0

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{x ^{2}}{e ^{x^{2}}} = (x^{2}) e^{- x^{2}}

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Simplify

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

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} ( 2 e ^{x^{2}} x ^{2} + e ^{x^{2}} )}

x \to \infty lim (1) = 1

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

= \frac{1}{\infty}

Apply Infinity Property:

\frac{c}{\infty} = 0

= 0

n = 0 \sum \infty \frac{1 ^{n}}{3 ^{n}}

\int - \frac{4 x + 10}{x ^{3}} d x

x \to 0 - lim (\frac{4}{tan ( x )})

\int e^{- s t} \cdot t d t

9 arcsec (x)

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