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Beliebt Rechnen >

limit as x approaches infinity of \sqrt[3]{x^2}*e^{-x}

Lösung

x \to \infty lim (3 x^{2} \cdot e^{- x})

Lösung

0

Schritte zur Lösung

x \to \infty lim (3 x^{2} e^{- x})

3 x^{2} = x^{\frac{2}{3}}

= x \to \infty lim (e^{- x} x^{\frac{2}{3}})

Umformung für L'Hopital

= x \to \infty lim (\frac{x ^{\frac{2}{3}}}{\frac{1}{e ^{- x}}})

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{\frac{2}{3 3 x}}{e ^{x}})

Wende Bruchregel an:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= x \to \infty lim (\frac{2}{3 e ^{x} 3 x})

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

= \frac{2}{3} \cdot x \to \infty lim (\frac{1}{e ^{x} 3 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

mit Einschränkung des unbestimmten Ausdrucks

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

x \to \infty lim (1) = 1

x \to \infty lim (e^{x} 3 x) = \infty

= \frac{2}{3} \cdot \frac{1}{\infty}

Vereinfache

\frac{2}{3} \cdot \frac{1}{\infty} : 0

= 0

Graph

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