Step 1: Interpreting the limit.
The expression \[ \lim_{x \to 0} \frac{\sin x}{x} \] represents the value that the ratio $\frac{\sin x}{x}$ tends to as $x$ approaches zero from both the left and right sides.
Step 2: Behavior near zero.
As $x \to 0$, \[ \sin x \to 0 \quad \text{and} \quad x \to 0 \] Thus, the expression takes the indeterminate form $\frac{0}{0}$, indicating that direct substitution is not sufficient and the limit must be evaluated properly.
Step 3: Applying a standard trigonometric limit.
A well-known fundamental result in calculus is: \[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \] This limit can be derived using geometric reasoning based on the unit circle or by comparing areas of triangles and circular sectors.
Step 4: Intuitive understanding.
When $x$ is very small (in radians), $\sin x$ is approximately equal to $x$. Therefore, the ratio $\frac{\sin x}{x}$ approaches $1$ as $x$ becomes closer to zero.
Step 5: Checking the given options.
(A) $0$: Incorrect, since $\sin x$ and $x$ approach zero at nearly the same rate.
(B) $1$: Correct, because the ratio tends to $1$ as $x \to 0$.
(C) $\infty$: Incorrect, the expression remains finite near zero.
(D) Does not exist: Incorrect, as both left-hand and right-hand limits are equal.
Step 6: Final result.
Since $\frac{\sin x}{x}$ approaches the same value from both sides as $x \to 0$, the value of the limit is 1.