\[\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{{\sin {\text{x}}}}{{\text{x}}}\] is A. indeterminate B. 0 C. 1 D. \[\infty \]

” correct=”option3″]

The correct answer is $\boxed{\text{indeterminate}}$.

The limit $\lim_{x\to 0}\frac{\sin x}{x}$ is indeterminate because as $x$ approaches 0, the ratio $\frac{\sin x}{x}$ approaches 1. However, the limit does not actually equal 1, because $\sin x$ is not always equal to $x$ when $x$ is close to 0. For example, when $x=0.1$, $\sin x=0.09959304$, and when $x=-0.1$, $\sin x=-0.09959304$. Therefore, the limit $\lim_{x\to 0}\frac{\sin x}{x}$ does not exist.

Option A is incorrect because the limit is not equal to 0. Option B is incorrect because the limit is not equal to 1. Option C is incorrect because the limit is not equal to $\infty$.