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

Lösung

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

0

Schritte zur Lösung

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

Umformung für L'Hopital

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

Wende das L'Hopital Theorem an

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

Vereinfache

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

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

Wende das L'Hopital Theorem an

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

Setze den Wert

x = 1

ein

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

Vereinfache

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

= 0

x \to \frac{π}{2} lim ((1 + cos (x))^{3 s e c (x)})

x \to 0 + lim (x cos (\frac{1}{x}))

x \to 1 lim (\frac{x ^{2}}{ln ( x )})

x \to 0 lim (\frac{x + cos ( x )}{x})

x \to 0 + lim ((x + sin (x))^{x})