Find the limit. lim _(xarrow 0)(sin^2(2x))/(sin^2)(9x)=

To find the limit of the given expression, we can use the trigonometric identity $\lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)} = \frac{a}{b}$, where $a$ and $b$ are constants.<br /><br />In this case, we have $a = 2$ and $b = 9$. Applying the identity, we get:<br /><br />$\lim_{x \to 0} \frac{\sin^2(2x)}{\sin^2(9x)} = \left(\frac{2}{9}\right)^2 = \frac{4}{81}$<br /><br />Therefore, the limit of the given expression is $\frac{4}{81}$.