What is the limit as x approaches-infinity of x^8e^x ?

The limit as x approaches-infinity of x^8e^x is 0

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limit as x approaches-infinity of x^8e^x

x \to - \infty lim (x^{8} e^{x})

0

Solution steps

x \to - \infty lim (x^{8} e^{x})

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{8 x ^{7}}{- e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{56 x ^{6}}{e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{336 x ^{5}}{- e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{1680 x ^{4}}{e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{6720 x ^{3}}{- e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{20160 x ^{2}}{e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{40320 x}{- e ^{- x}})

Apply L'Hopital's Rule

= x \to - \infty lim (\frac{40320}{e ^{- x}})

Simplify

\frac{40320}{e ^{- x}} : 40320 e^{x}

= x \to - \infty lim (40320 e^{x})

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

= 40320 \cdot x \to - \infty lim (e^{x})

Apply the common limit:

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

= 40320 \cdot 0

Simplify

= 0

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y^{^{'}} (x^{2} + 1) + 5 x (y - 1) = 0, y (0) = 7

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What is the limit as x approaches-infinity of x^8e^x ?
The limit as x approaches-infinity of x^8e^x is 0