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. Write the limit of $f(x)$ as $x\to 0$: ${\lim }_{x\to 0}\frac{\sin x}{x}$.
2. Recall the standard limit ${\lim }_{x\to 0}\frac{\sin x}{x}=1$.
3. Evaluate $f(0)$ from the definition: $f(0)=1$.
4. Since the limit equals the function value, $f$ is continuous at $x=0$.
5. Because the original expression $\frac{\sin x}{x}$ is undefined at $x=0$, the continuity is achieved by defining $f(0)=1$, which is a removable discontinuity that has been removed.

Concept tested: Continuity of piecewise function at a point

Published 7 September 2026 · How this solution was prepared and checked