Question

The limit $\displaystyle \lim_{x \to 0} \frac{\sin x}{x}$ is equal to:

• A. $0$
• B. $1$
• C. $\infty$
• D. Does not exist

Hint:

This limit is the foundation of derivatives of trigonometric functions, especially $\frac{d}{dx}(\sin x) = \cos x$.

Answer 1

The Correct Option is B

Step 1: Understanding the meaning of the limit.
The expression \[ \lim_{x \to 0} \frac{\sin x}{x} \] asks for the value that the ratio $\frac{\sin x}{x}$ approaches as $x$ gets very close to zero from both the positive and negative sides.
Step 2: Behavior of numerator and denominator near zero.
As $x \to 0$, \[ \sin x \to 0 \quad \text{and} \quad x \to 0 \] So the expression is of the indeterminate form $\frac{0}{0}$, which means we must evaluate the limit carefully rather than substituting directly.
Step 3: Using a fundamental trigonometric identity.
One of the most important standard limits in calculus is: \[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \] This result is obtained using geometric arguments involving the unit circle or by comparing areas of sectors and triangles.
Step 4: Conceptual explanation.
For very small values of $x$ (measured in radians), the value of $\sin x$ becomes almost equal to $x$. Hence, the ratio $\frac{\sin x}{x}$ becomes closer and closer to $1$ as $x$ approaches zero.
Step 5: Analysis of the given options.
(A) $0$: Incorrect, because $\sin x$ decreases at the same rate as $x$, not faster.
(B) $1$: Correct — the ratio approaches $1$ as $x \to 0$.
(C) $\infty$: Incorrect, the expression remains finite near zero.
(D) Does not exist: Incorrect, since the left-hand and right-hand limits are equal.
Step 6: Final conclusion.
Since the ratio $\frac{\sin x}{x}$ approaches $1$ from both sides as $x \to 0$, the value of the limit is 1.