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Beliebt Rechnen >

limit as x approaches infinity of ((e^{(15)/x}-1))/((15)/x)

Lösung

x \to \infty lim \frac{( e ^{\frac{15}{x}} - 1 )}{\frac{15}{x}}

Lösung

1

Schritte zur Lösung

x \to \infty lim (\frac{e ^{\frac{15}{x}} - 1}{\frac{15}{x}})

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

= \frac{1}{15} \cdot x \to \infty lim (\frac{e ^{\frac{15}{x}} - 1}{\frac{1}{x}})

Vereinfache

= \frac{1}{15} \cdot x \to \infty lim (x (e^{\frac{15}{x}} - 1))

Umformung für L'Hopital

= \frac{1}{15} \cdot x \to \infty lim (\frac{- 1 + e ^{\frac{15}{x}}}{\frac{1}{x}})

Wende das L'Hopital Theorem an

= \frac{1}{15} \cdot x \to \infty lim \frac{- \frac{15 e ^{\frac{15}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}}

Vereinfache

\frac{- \frac{15 e ^{\frac{15}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}} : 15 e^{\frac{15}{x}}

= \frac{1}{15} \cdot x \to \infty lim (15 e^{\frac{15}{x}})

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

= \frac{1}{15} \cdot 15 \cdot x \to \infty lim (e^{\frac{15}{x}})

Wende die Kettenregel für Grenzen an:

1

= \frac{1}{15} \cdot 15 \cdot 1

Vereinfache

\frac{1}{15} \cdot 15 \cdot 1 : 1

= 1

Graph

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