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

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

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

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

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

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

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= \frac{2}{3}

x \to 0 + lim (x cos (x))

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

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

x \to \infty lim (3 (x)^{2})

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

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