limit as x approaches 0+of ln(x)ln(1-x)

Lösung

x \to 0 + lim (ln (x) ln (1 - x))

0

Schritte zur Lösung

x \to 0 + lim (ln (x) ln (1 - x))

Umformung für L'Hopital

= x \to 0 + lim (\frac{ln ( x )}{\frac{1}{l n ( 1 - x )}})

Wende das L'Hopital Theorem an

= x \to 0 + lim (\frac{\frac{1}{x}}{\frac{1}{l n ^{2} ( 1 - x ) ( 1 - x )}})

Vereinfache

\frac{\frac{1}{x}}{\frac{1}{l n ^{2} ( 1 - x ) ( 1 - x )}} : \frac{ln ^{2} ( 1 - x ) ( 1 - x )}{x}

= x \to 0 + lim (\frac{ln ^{2} ( 1 - x ) ( 1 - x )}{x})

Wende das L'Hopital Theorem an

= x \to 0 + lim (\frac{- 2 ln ( 1 - x ) - ln ^{2} ( 1 - x )}{1})

Setze den Wert

x = 0

ein

= \frac{- 2 ln ( 1 - 0 ) - ln ^{2} ( 1 - 0 )}{1}

Vereinfache

\frac{- 2 ln ( 1 - 0 ) - ln ^{2} ( 1 - 0 )}{1} : 0

= 0

x \to 2 lim (\frac{5}{4 x - 8})

x \to - 1 lim (- x + 8)

x \to 0 lim ((1 + x^{2})^{3})

x \to 1 lim (\frac{1}{x} + ln (x))

x \to 1 lim (- x^{2} + 4 x - 5)