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límite cuando x tiende a infinity de (x^{99})/(e^x)

Solução

x \to \infty lim (\frac{x ^{99}}{e ^{x}})

0

Passos da solução

x \to \infty lim (\frac{x ^{99}}{e ^{x}})

Aplicar as propriedades dos expoentes:

\frac{1}{a ^{b}} = a^{- b}

\frac{x ^{99}}{e ^{x}} = (x^{99}) e^{- x}

= x \to \infty lim (x^{99} e^{- x})

Reescreva seguindo a regra de L'Hopital

= x \to \infty lim (\frac{e ^{- x}}{\frac{1}{x ^{99}}})

Aplicar a regra de L'Hopital

= x \to \infty lim (\frac{- e ^{- x}}{- \frac{99}{x ^{100}}})

Simplificar

\frac{- e ^{- x}}{- \frac{99}{x ^{100}}} : \frac{1}{99} e^{- x} x^{100}

= x \to \infty lim (\frac{1}{99} e^{- x} x^{100})

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

= \frac{1}{99} \cdot x \to \infty lim (e^{- x} x^{100})

Simplificar

e^{- x} : \frac{1}{e ^{x}}

= \frac{1}{99} \cdot x \to \infty lim (\frac{1}{e ^{x}} x^{100})

Simplificar

\frac{1}{e ^{x}} x^{100} : \frac{x ^{100}}{e ^{x}}

= \frac{1}{99} \cdot x \to \infty lim (\frac{x ^{100}}{e ^{x}})

Aplicar a regra de L'Hopital

= \frac{1}{99} \cdot x \to \infty lim (\frac{100 !}{e ^{x}})

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

= \frac{1}{99} \cdot 100! \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

Com exceção da forma indeterminada

= \frac{1}{99} \cdot 100! \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

= \frac{1}{99} \cdot 100! \cdot \frac{1}{\infty}

Simplificar

\frac{1}{99} \cdot 100! \cdot \frac{1}{\infty} : 0

= 0

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