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

Lösung

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

\infty

Schritte zur Lösung

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

Umformung für L'Hopital

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

Wende das L'Hopital Theorem an

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

Vereinfache

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

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

Wende die Kettenregel für Grenzen an:

\infty

= \infty

x \to 0 lim ((1 + 4 x)^{- \frac{3}{x}})

x \to 0 lim (1^{\frac{l n ( 6 )}{x}})

x \to - 4 lim (∣ x + 4 ∣ - 2 x)

x \to 0 lim (\frac{( 4 x ^{2} )}{e ^{4 x} - 4 x - 1})

x \to 2 lim (\frac{3 x}{x ^{2} - 4})