What is the limit as x approaches infinity of ((x^{12}))/(e^x) ?

The limit as x approaches infinity of ((x^{12}))/(e^x) is 0

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limit as x approaches infinity of ((x^{12}))/(e^x)

x \to \infty lim (\frac{( x ^{12} )}{e ^{x}})

0

Solution steps

x \to \infty lim (\frac{x ^{12}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{12 x ^{11}}{e ^{x}})

= x \to \infty lim (\frac{12 x ^{11}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{132 x ^{10}}{e ^{x}})

= x \to \infty lim (\frac{132 x ^{10}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{1320 x ^{9}}{e ^{x}})

= x \to \infty lim (\frac{1320 x ^{9}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{11880 x ^{8}}{e ^{x}})

= x \to \infty lim (\frac{11880 x ^{8}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{95040 x ^{7}}{e ^{x}})

= x \to \infty lim (\frac{95040 x ^{7}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{665280 x ^{6}}{e ^{x}})

= x \to \infty lim (\frac{665280 x ^{6}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{3991680 x ^{5}}{e ^{x}})

= x \to \infty lim (\frac{3991680 x ^{5}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{19958400 x ^{4}}{e ^{x}})

= x \to \infty lim (\frac{19958400 x ^{4}}{e ^{x}})

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

= 19958400 \cdot x \to \infty lim (\frac{x ^{4}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{4 x ^{3}}{e ^{x}})

= 19958400 \cdot x \to \infty lim (\frac{4 x ^{3}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{12 x ^{2}}{e ^{x}})

= 19958400 \cdot x \to \infty lim (\frac{12 x ^{2}}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{24 x}{e ^{x}})

= 19958400 \cdot x \to \infty lim (\frac{24 x}{e ^{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{24}{e ^{x}})

= 19958400 \cdot x \to \infty lim (\frac{24}{e ^{x}})

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

= 19958400 \cdot 24 \cdot x \to \infty lim (\frac{1}{e ^{x}})

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

= 19958400 \cdot 24 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( e ^{x} )}

x \to \infty lim (1) = 1

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

= 19958400 \cdot 24 \cdot \frac{1}{\infty}

Simplify

19958400 \cdot 24 \cdot \frac{1}{\infty} : 0

= 0

x \to 4 - lim (x - 4)

x \to π lim (\frac{cos ( x ) - 1}{4})

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

x \to 3 lim (- f (x))

x \to 4 - lim (\frac{8}{x - 4})

What is the limit as x approaches infinity of ((x^{12}))/(e^x) ?
The limit as x approaches infinity of ((x^{12}))/(e^x) is 0