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

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

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

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

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= \frac{2}{3} \cdot x \to 0 lim (\frac{arcsin ( x )}{x})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{2}{3} \cdot \frac{\frac{1}{1 - 0 ^{2}}}{1}

Simplify

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

= \frac{2}{3}

x \to - 2 lim (x^{2} + 2 x - 15)

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

x \to 0 lim (\frac{4 sin ( 3 x )}{x})

x \to \infty lim (1, + x)

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

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