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Beliebt Rechnen >

limit as x approaches 0 of (arcsin(2x)-2arcsin(x))/(x^3)

Lösung

x \to 0 lim (\frac{arcsin ( 2 x ) - 2 arcsin ( x )}{x ^{3}})

Lösung

1

Schritte zur Lösung

x \to 0 lim (\frac{arcsin ( 2 x ) - 2 arcsin ( x )}{x ^{3}})

Wende das L'Hopital Theorem an

= x \to 0 lim (\frac{\frac{2}{1 - 4 x ^{2}} - \frac{2}{1 - x ^{2}}}{3 x ^{2}})

Wende das L'Hopital Theorem an

= x \to 0 lim \frac{\frac{8 x}{( 1 - 4 x ^{2} ) ^{\frac{3}{2}}} - \frac{2 x}{( 1 - x ^{2} ) ^{\frac{3}{2}}}}{6 x}

Vereinfache

\frac{\frac{8 x}{( 1 - 4 x ^{2} ) ^{\frac{3}{2}}} - \frac{2 x}{( 1 - x ^{2} ) ^{\frac{3}{2}}}}{6 x} : \frac{4 ( - x ^{2} + 1 ) ^{\frac{3}{2}} - ( - 4 x ^{2} + 1 ) ^{\frac{3}{2}}}{3 ( - x ^{2} + 1 ) ^{\frac{3}{2}} ( - 4 x ^{2} + 1 ) ^{\frac{3}{2}}}

= x \to 0 lim \frac{4 ( - x ^{2} + 1 ) ^{\frac{3}{2}} - ( - 4 x ^{2} + 1 ) ^{\frac{3}{2}}}{3 ( - x ^{2} + 1 ) ^{\frac{3}{2}} ( - 4 x ^{2} + 1 ) ^{\frac{3}{2}}}

Setze den Wert

x = 0

ein

= \frac{4 ( - 0 ^{2} + 1 ) ^{\frac{3}{2}} - ( - 4 \cdot 0 ^{2} + 1 ) ^{\frac{3}{2}}}{3 ( - 0 ^{2} + 1 ) ^{\frac{3}{2}} ( - 4 \cdot 0 ^{2} + 1 ) ^{\frac{3}{2}}}

Vereinfache

\frac{4 ( - 0 ^{2} + 1 ) ^{\frac{3}{2}} - ( - 4 \cdot 0 ^{2} + 1 ) ^{\frac{3}{2}}}{3 ( - 0 ^{2} + 1 ) ^{\frac{3}{2}} ( - 4 \cdot 0 ^{2} + 1 ) ^{\frac{3}{2}}} : 1

= 1

Graph

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