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

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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

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

= 3 \cdot 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}} )})

= 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

x \to - 2 lim (- \frac{x + 1}{x ^{2} - 1})

9 \to 9 lim (6820 + \frac{81 - 9 ^{2}}{9 - 9})

x \to 2 lim (\frac{- 3}{x - 2})

y \to 3 lim (y^{2} - 2 y + 7)

x \to \infty lim (25)

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