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

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

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limit as x approaches 0 of (2x-arcsin(x))/(2x+arctan(x))

x \to 0 lim (\frac{2 x - arcsin ( x )}{2 x + arctan ( x )})

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

x \to 0 lim (\frac{2 x - arcsin ( x )}{2 x + arctan ( x )})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{2 - \frac{1}{1 - 0 ^{2}}}{2 + \frac{1}{0 ^{2} + 1}}

Simplify

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

= \frac{1}{3}

x \to 0 + lim (x^{2} ln (4 x^{2}))

x \to 2 lim (\frac{sin ( x - 2 )}{3 x - 6})

x \to 0 + lim (x^{2} csc^{2} (3 x))

x \to 0 + lim (x^{\frac{1}{f ( x )}})

x \to 0 - lim ((1 - e^{x})^{x})

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