What is the limit as x approaches-infinity of x^5e^x ?

The limit as x approaches-infinity of x^5e^x is 0

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limit as x approaches-infinity of x^5e^x

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

0

Solution steps

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

Rewrite for L'Hopital

= x \to - \infty lim (\frac{x ^{5}}{\frac{1}{e ^{x}}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{5 x ^{4}}{- e ^{- x}})

= x \to - \infty lim (\frac{5 x ^{4}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{20 x ^{3}}{e ^{- x}})

= x \to - \infty lim (\frac{20 x ^{3}}{e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{60 x ^{2}}{- e ^{- x}})

= x \to - \infty lim (\frac{60 x ^{2}}{- e ^{- x}})

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Simplify

\frac{120}{- e ^{- x}} : - 120 e^{x}

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

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

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

Apply the common limit:

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

= - 120 \cdot 0

Simplify

= 0

x \to - 3 lim (\frac{1}{( x + 3 ) ^{5}})

x \to 0 lim (\frac{5 x - 2}{x})

x \to 0 lim (- \frac{x}{sin ( x )})

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

x \to \infty lim (\frac{x}{π})

What is the limit as x approaches-infinity of x^5e^x ?
The limit as x approaches-infinity of x^5e^x is 0