CBSE 2025, Set 1

Question

Read the following statements: Assertion (A) and Reason (R). Select the correct answer.

**Assertion (A):** $f(x) = \begin{cases} x\sin\dfrac{1}{x}, & x 
eq 0 \ 0, & x = 0 \end{cases}$ is continuous at $x = 0$.

**Reason (R):** When $x 	o 0$, $\sin\dfrac{1}{x}$ is a finite value between $-1$ and $1$. A). Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). B). Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C). Assertion (A) is true, but Reason (R) is false. D). Assertion (A) is false, but Reason (R) is true.

Edvaya · Accepted Answer

The correct answer is Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).. **Assertion (A) is TRUE:** $$\lim_{x 	o 0} x\sin\frac{1}{x}$$ Since $-1 \leq \sin\dfrac{1}{x} \leq 1$ for all $x 
eq 0$: $$-|x| \leq x\sin\frac{1}{x} \leq |x|$$ By the Squeeze Theorem: $\lim_{x 	o 0} x\sin\dfrac{1}{x} = 0 = f(0)$. So $f$ is continuous at $x = 0$. ✓ **Reason (R) is TRUE:** $\sin\dfrac{1}{x}$ is bounded between $-1$ and $1$ for all $x 
eq 0$. ✓ **R correctly explains A:** The continuity follows precisely because $\sin(1/x)$ is bounded — multiplying a bounded quantity by $x 	o 0$ forces the limit to zero.

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Mathematics > Calculus > Continuity And Differentiability

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