limit as x approaches infinity of (ln^3(x))/(100\sqrt[3]{x)}

Lösung

0

Schritte zur Lösung

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

Rationalisiere den Nenner

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

Vereinfache

\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}}})

Wende das L'Hopital Theorem an

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

Vereinfache

\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}}})

Wende das L'Hopital Theorem an

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

Vereinfache

\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}}})

Wende das L'Hopital Theorem an

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

Vereinfache

\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}}})

Wende die Unendlichkeitseigenschaft an:

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

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

Vereinfache

= 0