limit as x approaches 0+of x^2(ln(x))^2

Lösung

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

0

Schritte zur Lösung

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

Wende Exponentenregel an:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{2} (ln (x))^{2} = (x ln (x))^{2}

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

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

mit Einschränkung des unbestimmten Ausdrucks

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

Umformung für L'Hopital

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

Wende das L'Hopital Theorem an

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

Vereinfache

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

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

Setze den Wert

x = 0

ein

= (- 0)^{2}

Vereinfache

= 0

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

x \to 2 lim (- 2 x + 4)

x \to 0 - lim (\frac{9 sin ( x )}{x})

θ \to \frac{π}{6} lim (\frac{sin ( 2 θ )}{sin ( θ ) tan ( θ )})

x \to \infty lim (40)