What is the limit as x approaches 0 of (arcsin(x)-arctan(x))/(x^3) ?

The limit as x approaches 0 of (arcsin(x)-arctan(x))/(x^3) is 1/2

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

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

\frac{1}{2}

Decimal

0.5

Solution steps

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Simplify

\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}}}

Plug in the value

x = 0

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

Simplify

\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}

What is the limit as x approaches 0 of (arcsin(x)-arctan(x))/(x^3) ?
The limit as x approaches 0 of (arcsin(x)-arctan(x))/(x^3) is 1/2