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

The limit as x approaches infinity of ((e^{x^5}))/(5-x^6) is -infinity

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

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

- \infty

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: x \to \infty lim \frac{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}} )}{- 720 x}

= x \to \infty lim \frac{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}} )}{- 720 x}

Apply L'Hopital's Rule

: x \to \infty lim \frac{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} )}{- 720}

= x \to \infty lim \frac{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} )}{- 720}

Simplify

\frac{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} )}{- 720} : - \frac{1}{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 \infty lim (- \frac{1}{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)

= - \frac{1}{144} \cdot 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})

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

With the exception of indeterminate form

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

x \to \infty lim (3125 e^{x^{5}} x^{24}) = \infty

x \to \infty lim (37500 e^{x^{5}} x^{19}) = \infty

x \to \infty lim (120000 e^{x^{5}} x^{14}) = \infty

x \to \infty lim (105000 e^{x^{5}} x^{9}) = \infty

x \to \infty lim (15120 e^{x^{5}} x^{4}) = \infty

= - \frac{1}{144} (\infty + \infty + \infty + \infty + \infty)

Simplify

- \frac{1}{144} (\infty + \infty + \infty + \infty + \infty) : - \infty

= - \infty

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