319. \( \lim _{x \rightarrow 0} \frac{2 \arcsin x}{3 x} \).

To evaluate the limit \( \lim _{x \rightarrow 0} \frac{2 \arcsin x}{3 x} \), we can first substitute \( x = 0 \) directly into the expression. However, this gives us an indeterminate form \( \frac{0}{0} \). Therefore, we can apply L'Hôpital's Rule, which is applicable in such cases. We differentiate the numerator and the denominator: 1. The derivative of the numerator \( 2 \arcsin x \) is: \[ \frac{d}{dx}(2 \arcsin x) = 2 \cdot \frac{1}{\sqrt{1 - x^2}}. \] 2. The derivative of the denominator \( 3x \) is: \[ \frac{d}{dx}(3x) = 3. \] Now we apply L'Hôpital's Rule: \[ \lim _{x \rightarrow 0} \frac{2 \arcsin x}{3 x} = \lim _{x \rightarrow 0} \frac{2 / \sqrt{1 - x^2}}{3}. \] Evaluating the limit as \( x \) approaches 0: \[ \frac{2}{3 \sqrt{1 - 0^2}} = \frac{2}{3 \cdot 1} = \frac{2}{3}. \] Thus, the limit is: \[ \boxed{\frac{2}{3}}. \]