Consider the function $f(x)=\{\begin{array}{ll}\frac{\sin x}{x},& x\ne 0\\ 1,& x=0\end{array}$. Which of the following statements about the continuity of $f$ at $x=0$ is correct?

Step-by-step solution

1. Recall the standard limit ${\lim }_{x\to 0}\frac{\sin x}{x}=1$.
2. Compute ${\lim }_{x\to 0}f(x)={\lim }_{x\to 0}\frac{\sin x}{x}=1$.
3. Evaluate the function value at $x=0$: $f(0)=1$ (given).
4. Since the limit equals the function value, $f$ is continuous at $0$.
5. Therefore option (A) is correct.

Concept tested: Continuity of piecewise trigonometric function

Published 16 September 2026 · How this solution was prepared and checked