What is the limit as x approaches-infinity of (2e^{-1/2 x})/(x^3) ?

The limit as x approaches-infinity of (2e^{-1/2 x})/(x^3) is -infinity

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limit as x approaches-infinity of (2e^{-1/2 x})/(x^3)

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

- \infty

Solution steps

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

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{\frac{1}{4} e ^{- \frac{x}{2}}}{6 x})

= 2 \cdot x \to - \infty lim (\frac{\frac{1}{4} e ^{- \frac{x}{2}}}{6 x})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{- \frac{1}{8} e ^{- \frac{x}{2}}}{6})

= 2 \cdot x \to - \infty lim (\frac{- \frac{1}{8} e ^{- \frac{x}{2}}}{6})

Simplify

\frac{- \frac{1}{8} e ^{- \frac{x}{2}}}{6} : - \frac{1}{48} e^{- \frac{x}{2}}

= 2 \cdot x \to - \infty lim (- \frac{1}{48} e^{- \frac{x}{2}})

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

= 2 (- \frac{1}{48} \cdot x \to - \infty lim (e^{- \frac{x}{2}}))

Apply the Limit Chain Rule:

\infty

= 2 (- \frac{1}{48} \cdot \infty)

Simplify

2 (- \frac{1}{48} \cdot \infty) : - \infty

= - \infty

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What is the limit as x approaches-infinity of (2e^{-1/2 x})/(x^3) ?
The limit as x approaches-infinity of (2e^{-1/2 x})/(x^3) is -infinity