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límite cuando x tiende a infinity de (ln^3(x))/(100\sqrt[3]{x)}

Solução

x \to \infty lim (\frac{ln ^{3} ( x )}{100 3 x})

0

Passos da solução

x \to \infty lim (\frac{ln ^{3} ( x )}{100 3 x})

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

= \frac{1}{100} \cdot x \to \infty lim (\frac{ln ^{3} ( x )}{3 x})

Racionalizar o denominador

\frac{ln ^{3} ( x )}{3 x} : \frac{x ^{\frac{2}{3}} ln ^{3} ( x )}{x}

= \frac{1}{100} \cdot x \to \infty lim (\frac{x ^{\frac{2}{3}} ln ^{3} ( x )}{x})

Simplificar

\frac{x ^{\frac{2}{3}} ln ^{3} ( x )}{x} : \frac{ln ^{3} ( x )}{x ^{\frac{1}{3}}}

= \frac{1}{100} \cdot x \to \infty lim (\frac{ln ^{3} ( x )}{x ^{\frac{1}{3}}})

Aplicar a regra de L'Hopital

= \frac{1}{100} \cdot x \to \infty lim (\frac{\frac{3 l n ^{2} ( x )}{x}}{\frac{1}{3 x ^{\frac{2}{3}}}})

Simplificar

\frac{\frac{3 l n ^{2} ( x )}{x}}{\frac{1}{3 x ^{\frac{2}{3}}}} : \frac{9 ln ^{2} ( x )}{x ^{\frac{1}{3}}}

= \frac{1}{100} \cdot x \to \infty lim (\frac{9 ln ^{2} ( x )}{x ^{\frac{1}{3}}})

Aplicar a regra de L'Hopital

= \frac{1}{100} \cdot x \to \infty lim (\frac{\frac{18 l n ( x )}{x}}{\frac{1}{3 x ^{\frac{2}{3}}}})

Simplificar

\frac{\frac{18 l n ( x )}{x}}{\frac{1}{3 x ^{\frac{2}{3}}}} : \frac{54 ln ( x )}{x ^{\frac{1}{3}}}

= \frac{1}{100} \cdot x \to \infty lim (\frac{54 ln ( x )}{x ^{\frac{1}{3}}})

Aplicar a regra de L'Hopital

= \frac{1}{100} \cdot x \to \infty lim (\frac{\frac{54}{x}}{\frac{1}{3 x ^{\frac{2}{3}}}})

Simplificar

\frac{\frac{54}{x}}{\frac{1}{3 x ^{\frac{2}{3}}}} : \frac{162}{x ^{\frac{1}{3}}}

= \frac{1}{100} \cdot x \to \infty lim (\frac{162}{x ^{\frac{1}{3}}})

Aplicar as propriedades dos limites infinitos:

x \to \infty lim (\frac{c}{x ^{a}}) = 0

= \frac{1}{100} \cdot 0

Simplificar

= 0

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