What is the limit as x approaches-infinity of x^6e^x ?

The limit as x approaches-infinity of x^6e^x is 0

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limit as x approaches-infinity of x^6e^x

x \to - \infty lim (x^{6} e^{x})

0

Solution steps

x \to - \infty lim (x^{6} e^{x})

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{6 x ^{5}}{- e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{30 x ^{4}}{e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{120 x ^{3}}{- e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{360 x ^{2}}{e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{720 x}{- e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{720}{e ^{- x}})

Simplify

\frac{720}{e ^{- x}} : 720 e^{x}

= x \to - \infty lim (720 e^{x})

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

= 720 \cdot x \to - \infty lim (e^{x})

Apply the common limit:

x \to - \infty lim (e^{x}) = 0

= 720 \cdot 0

Simplify

= 0

What is the limit as x approaches-infinity of x^6e^x ?
The limit as x approaches-infinity of x^6e^x is 0