What is the limit as x approaches infinity of x^6e^{-x^5} ?

The limit as x approaches infinity of x^6e^{-x^5} is 0

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limit as x approaches infinity of x^6e^{-x^5}

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

0

Solution steps

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

Rewrite for L'Hopital

= x \to \infty lim (\frac{x ^{6}}{\frac{1}{e ^{- x^{5}}}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{6 x ^{5}}{e ^{x^{5}} \cdot 5 x ^{4}})

= x \to \infty lim (\frac{6 x ^{5}}{e ^{x^{5}} \cdot 5 x ^{4}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{30 x ^{4}}{5 ( 5 e ^{x^{5}} x ^{8} + 4 e ^{x^{5}} x ^{3} )})

= x \to \infty lim (\frac{30 x ^{4}}{5 ( 5 e ^{x^{5}} x ^{8} + 4 e ^{x^{5}} x ^{3} )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{120 x ^{3}}{5 ( 25 e ^{x^{5}} x ^{12} + 60 e ^{x^{5}} x ^{7} + 12 e ^{x^{5}} x ^{2} )})

= x \to \infty lim (\frac{120 x ^{3}}{5 ( 25 e ^{x^{5}} x ^{12} + 60 e ^{x^{5}} x ^{7} + 12 e ^{x^{5}} x ^{2} )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{360 x ^{2}}{5 ( 125 e ^{x^{5}} x ^{16} + 600 e ^{x^{5}} x ^{11} + 480 e ^{x^{5}} x ^{6} + 24 e ^{x^{5}} x )})

= x \to \infty lim (\frac{360 x ^{2}}{5 ( 125 e ^{x^{5}} x ^{16} + 600 e ^{x^{5}} x ^{11} + 480 e ^{x^{5}} x ^{6} + 24 e ^{x^{5}} x )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{720 x}{5 ( 625 e ^{x^{5}} x ^{20} + 5000 e ^{x^{5}} x ^{15} + 9000 e ^{x^{5}} x ^{10} + 3000 e ^{x^{5}} x ^{5} + 24 e ^{x^{5}} )})

= x \to \infty lim (\frac{720 x}{5 ( 625 e ^{x^{5}} x ^{20} + 5000 e ^{x^{5}} x ^{15} + 9000 e ^{x^{5}} x ^{10} + 3000 e ^{x^{5}} x ^{5} + 24 e ^{x^{5}} )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{720}{5 ( 3125 e ^{x^{5}} x ^{24} + 37500 e ^{x^{5}} x ^{19} + 120000 e ^{x^{5}} x ^{14} + 105000 e ^{x^{5}} x ^{9} + 15120 e ^{x^{5}} x ^{4} )})

= x \to \infty lim (\frac{720}{5 ( 3125 e ^{x^{5}} x ^{24} + 37500 e ^{x^{5}} x ^{19} + 120000 e ^{x^{5}} x ^{14} + 105000 e ^{x^{5}} x ^{9} + 15120 e ^{x^{5}} x ^{4} )})

Simplify

= x \to \infty lim (\frac{144}{3125 e ^{x^{5}} x ^{24} + 37500 e ^{x^{5}} x ^{19} + 120000 e ^{x^{5}} x ^{14} + 105000 e ^{x^{5}} x ^{9} + 15120 e ^{x^{5}} x ^{4}})

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

= 144 \cdot x \to \infty lim (\frac{1}{3125 e ^{x^{5}} x ^{24} + 37500 e ^{x^{5}} x ^{19} + 120000 e ^{x^{5}} x ^{14} + 105000 e ^{x^{5}} x ^{9} + 15120 e ^{x^{5}} x ^{4}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= 144 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 3125 e ^{x^{5}} x ^{24} + 37500 e ^{x^{5}} x ^{19} + 120000 e ^{x^{5}} x ^{14} + 105000 e ^{x^{5}} x ^{9} + 15120 e ^{x^{5}} x ^{4} )}

x \to \infty lim (1) = 1

x \to \infty lim (3125 e^{x^{5}} x^{24} + 37500 e^{x^{5}} x^{19} + 120000 e^{x^{5}} x^{14} + 105000 e^{x^{5}} x^{9} + 15120 e^{x^{5}} x^{4}) = \infty

= 144 \cdot \frac{1}{\infty}

Simplify

144 \cdot \frac{1}{\infty} : 0

= 0

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\frac{\partial}{\partial x} ((x + z - y)^{2})

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What is the limit as x approaches infinity of x^6e^{-x^5} ?
The limit as x approaches infinity of x^6e^{-x^5} is 0