What is the limit as x approaches-infinity of x^9e^x ?

The limit as x approaches-infinity of x^9e^x is 0

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limit as x approaches-infinity of x^9e^x

x \to - \infty lim (x^{9} e^{x})

0

Solution steps

x \to - \infty lim (x^{9} e^{x})

Rewrite for L'Hopital

= x \to - \infty lim (\frac{x ^{9}}{\frac{1}{e ^{x}}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{9 x ^{8}}{- e ^{- x}})

= x \to - \infty lim (\frac{9 x ^{8}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{72 x ^{7}}{e ^{- x}})

= x \to - \infty lim (\frac{72 x ^{7}}{e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{504 x ^{6}}{- e ^{- x}})

= x \to - \infty lim (\frac{504 x ^{6}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{3024 x ^{5}}{e ^{- x}})

= x \to - \infty lim (\frac{3024 x ^{5}}{e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{15120 x ^{4}}{- e ^{- x}})

= x \to - \infty lim (\frac{15120 x ^{4}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{60480 x ^{3}}{e ^{- x}})

= x \to - \infty lim (\frac{60480 x ^{3}}{e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{181440 x ^{2}}{- e ^{- x}})

= x \to - \infty lim (\frac{181440 x ^{2}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{362880 x}{e ^{- x}})

= x \to - \infty lim (\frac{362880 x}{e ^{- x}})

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

= 362880 \cdot x \to - \infty lim (\frac{x}{e ^{- x}})

Simplify

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

= 362880 \cdot x \to - \infty lim (e^{x} x)

Rewrite for L'Hopital

= 362880 \cdot x \to - \infty lim (\frac{x}{\frac{1}{e ^{x}}})

Apply L'Hopital's Rule

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

= 362880 \cdot x \to - \infty lim (\frac{1}{- e ^{- x}})

Simplify

\frac{1}{- e ^{- x}} : - e^{x}

= 362880 \cdot x \to - \infty lim (- e^{x})

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

= 362880 (- x \to - \infty lim (e^{x}))

Apply the common limit:

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

= 362880 (- 0)

Simplify

= 0

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

x \to 0 lim ((1 + 1 x)^{\frac{1}{x}})

t \to 0 + lim (\frac{sin ( 2 t )}{t})

x \to 0 lim (x sin (\frac{1}{x + y}))

x \to 0 lim (1 0^{x} + 2)

What is the limit as x approaches-infinity of x^9e^x ?
The limit as x approaches-infinity of x^9e^x is 0