limit as x approaches 0 of (arcsin(x)-arctan(x))/(x^3)

Lösung

x \to 0 lim (\frac{arcsin ( x ) - arctan ( x )}{x ^{3}})

\frac{1}{2}

Dezimale

0.5

Schritte zur Lösung

x \to 0 lim (\frac{arcsin ( x ) - arctan ( x )}{x ^{3}})

Wende das L'Hopital Theorem an

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

Wende das L'Hopital Theorem an

= x \to 0 lim \frac{\frac{x}{( 1 - x ^{2} ) ^{\frac{3}{2}}} + \frac{2 x}{( x ^{2} + 1 ) ^{2}}}{6 x}

Vereinfache

\frac{\frac{x}{( 1 - x ^{2} ) ^{\frac{3}{2}}} + \frac{2 x}{( x ^{2} + 1 ) ^{2}}}{6 x} : \frac{( x ^{2} + 1 ) ^{2} + 2 ( - x ^{2} + 1 ) ^{\frac{3}{2}}}{6 ( x ^{2} + 1 ) ^{2} ( - x ^{2} + 1 ) ^{\frac{3}{2}}}

= x \to 0 lim \frac{( x ^{2} + 1 ) ^{2} + 2 ( - x ^{2} + 1 ) ^{\frac{3}{2}}}{6 ( x ^{2} + 1 ) ^{2} ( - x ^{2} + 1 ) ^{\frac{3}{2}}}

Setze den Wert

x = 0

ein

= \frac{( 0 ^{2} + 1 ) ^{2} + 2 ( - 0 ^{2} + 1 ) ^{\frac{3}{2}}}{6 ( 0 ^{2} + 1 ) ^{2} ( - 0 ^{2} + 1 ) ^{\frac{3}{2}}}

Vereinfache

\frac{( 0 ^{2} + 1 ) ^{2} + 2 ( - 0 ^{2} + 1 ) ^{\frac{3}{2}}}{6 ( 0 ^{2} + 1 ) ^{2} ( - 0 ^{2} + 1 ) ^{\frac{3}{2}}} : \frac{1}{2}

= \frac{1}{2}

x \to a lim (f (x) (x)^{n})

x \to - 6 lim (10 x + 1)

x \to 3 + lim (3 - x)

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

x \to 0 lim (x + 0.12 (0))