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Beliebt Rechnen >

limit as x approaches 0-of (e^{1/x})/x

Lösung

x \to 0 - lim (\frac{e ^{\frac{1}{x}}}{x})

Lösung

0

Schritte zur Lösung

x \to 0 - lim (\frac{e ^{\frac{1}{x}}}{x})

\frac{e ^{\frac{1}{x}}}{x} = e^{\frac{1}{x}} \frac{1}{x}

= x \to 0 - lim (e^{\frac{1}{x}} \frac{1}{x})

Umformung für L'Hopital

= x \to 0 - lim (\frac{\frac{1}{x}}{\frac{1}{e ^{\frac{1}{x}}}})

Wende das L'Hopital Theorem an

= x \to 0 - lim \frac{- \frac{1}{x ^{2}}}{\frac{e ^{- \frac{1}{x}}}{x ^{2}}}

Vereinfache

\frac{- \frac{1}{x ^{2}}}{\frac{e ^{- \frac{1}{x}}}{x ^{2}}} : - e^{\frac{1}{x}}

= x \to 0 - lim (- e^{\frac{1}{x}})

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

= - x \to 0 - lim (e^{\frac{1}{x}})

Wende die Kettenregel für Grenzen an:

0

= - 0

Vereinfache

= 0

Graph

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