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

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

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

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

1

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: 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}

= 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}

Simplify

\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}}}

Plug in the value

x = 0

= \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}}}

Simplify

\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

f (x) = x^{2}

x \to 3 lim (\frac{x - 2}{x + 5})

2 x^{2} y \frac{d y}{d x} + 1 + 2 x y^{2} = 0

\frac{d ^{3}}{d x ^{3}} (cot (2 x) + 3 x^{2})

\frac{d}{d x} (x^{4} - 8 x^{3} + 4 x^{2} + 48 x)

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