What is the limit as x approaches-infinity of x^7e^x ?

The limit as x approaches-infinity of x^7e^x is 0

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limit as x approaches-infinity of x^7e^x

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

0

Solution steps

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

Rewrite for L'Hopital

= x \to - \infty lim (\frac{x ^{7}}{\frac{1}{e ^{x}}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{7 x ^{6}}{- e ^{- x}})

= x \to - \infty lim (\frac{7 x ^{6}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{42 x ^{5}}{e ^{- x}})

= x \to - \infty lim (\frac{42 x ^{5}}{e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{210 x ^{4}}{- e ^{- x}})

= x \to - \infty lim (\frac{210 x ^{4}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{840 x ^{3}}{e ^{- x}})

= x \to - \infty lim (\frac{840 x ^{3}}{e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{2520 x ^{2}}{- e ^{- x}})

= x \to - \infty lim (\frac{2520 x ^{2}}{- e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{5040 x}{e ^{- x}})

= x \to - \infty lim (\frac{5040 x}{e ^{- x}})

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{5040}{- e ^{- x}})

= x \to - \infty lim (\frac{5040}{- e ^{- x}})

Simplify

\frac{5040}{- e ^{- x}} : - 5040 e^{x}

= x \to - \infty lim (- 5040 e^{x})

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

= - 5040 \cdot x \to - \infty lim (e^{x})

Apply the common limit:

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

= - 5040 \cdot 0

Simplify

= 0

x \to \infty lim (e^{x} + 2)

x \to 0 - lim (sin (x) ln (x))

x \to - 3 lim (x^{2} - 4)

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

x \to \frac{π}{2} lim (\frac{cos ( x )}{x})

What is the limit as x approaches-infinity of x^7e^x ?
The limit as x approaches-infinity of x^7e^x is 0