What is the limit as x approaches infinity of (3x^2)/(e^{x^2)} ?

The limit as x approaches infinity of (3x^2)/(e^{x^2)} is 0

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

limit as x approaches infinity of (3x^2)/(e^{x^2)}

x \to \infty lim (\frac{3 x ^{2}}{e ^{x^{2}}})

0

Solution steps

x \to \infty lim (\frac{3 x ^{2}}{e ^{x^{2}}})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 3 \cdot 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}}

= 3 \cdot x \to \infty lim (x^{2} e^{- x^{2}})

Rewrite for L'Hopital

= 3 \cdot x \to \infty lim (\frac{x ^{2}}{\frac{1}{e ^{- x^{2}}}})

Apply L'Hopital's Rule

= 3 \cdot x \to \infty lim (\frac{2 x}{e ^{x^{2}} \cdot 2 x})

Apply L'Hopital's Rule

= 3 \cdot x \to \infty lim (\frac{2}{2 ( 2 e ^{x^{2}} x ^{2} + e ^{x^{2}} )})

Simplify

= 3 \cdot 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

= 3 \cdot \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

= 3 \cdot \frac{1}{\infty}

Simplify

3 \cdot \frac{1}{\infty} : 0

= 0

What is the limit as x approaches infinity of (3x^2)/(e^{x^2)} ?
The limit as x approaches infinity of (3x^2)/(e^{x^2)} is 0