What is the limit as x approaches infinity+of (x^3)/(1e^{x/4)} ?

The limit as x approaches infinity+of (x^3)/(1e^{x/4)} is 0

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limit as x approaches infinity+of (x^3)/(1e^{x/4)}

x \to \infty + lim (\frac{x ^{3}}{1 e ^{\frac{x}{4}}})

0

Solution steps

x \to \infty lim (\frac{x ^{3}}{1 \cdot e ^{\frac{x}{4}}})

Simplify

= x \to \infty lim (\frac{x ^{3}}{e ^{\frac{x}{4}}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{3 x ^{2}}{e ^{\frac{x}{4}} \frac{1}{4}})

= x \to \infty lim (\frac{3 x ^{2}}{e ^{\frac{x}{4}} \frac{1}{4}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{6 x}{\frac{1}{16} e ^{\frac{x}{4}}})

= x \to \infty lim (\frac{6 x}{\frac{1}{16} e ^{\frac{x}{4}}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{6}{\frac{1}{64} e ^{\frac{x}{4}}})

= x \to \infty lim (\frac{6}{\frac{1}{64} e ^{\frac{x}{4}}})

Simplify

\frac{6}{\frac{1}{64} e ^{\frac{x}{4}}} : \frac{384}{e ^{\frac{x}{4}}}

= x \to \infty lim (\frac{384}{e ^{\frac{x}{4}}})

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

= 384 \cdot x \to \infty lim (\frac{1}{e ^{\frac{x}{4}}})

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

= 384 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( e ^{\frac{x}{4}} )}

x \to \infty lim (1) = 1

x \to \infty lim (e^{\frac{x}{4}}) = \infty

= 384 \cdot \frac{1}{\infty}

Simplify

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

= 0

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What is the limit as x approaches infinity+of (x^3)/(1e^{x/4)} ?
The limit as x approaches infinity+of (x^3)/(1e^{x/4)} is 0