Class CBSE Class 12 Mathematics Continuity and Differentiability Q #1334
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2024 AISSCE(Board Exam) VSA
Verify whether the function f defined by $f(x)=\begin{cases}x~sin(\frac{1}{x}),x\ne0\\ 0&,x=0\end{cases}$ is continuous at $x=0$ or not.
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Detailed Solution

Step 1: Check the left-hand limit (LHL) at x=0

We need to find the limit of $f(x)$ as $x$ approaches 0 from the left side. $$LHL = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x \sin(\frac{1}{x})$$ Since $-1 \le \sin(\frac{1}{x}) \le 1$, we have $-|x| \le x \sin(\frac{1}{x}) \le |x|$. As $x \to 0^-$, $|x| \to 0$. Therefore, by the Squeeze Theorem, $$LHL = \lim_{x \to 0^-} x \sin(\frac{1}{x}) = 0$$

Step 2: Check the right-hand limit (RHL) at x=0

We need to find the limit of $f(x)$ as $x$ approaches 0 from the right side. $$RHL = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x \sin(\frac{1}{x})$$ Since $-1 \le \sin(\frac{1}{x}) \le 1$, we have $-|x| \le x \sin(\frac{1}{x}) \le |x|$. As $x \to 0^+$, $|x| \to 0$. Therefore, by the Squeeze Theorem, $$RHL = \lim_{x \to 0^+} x \sin(\frac{1}{x}) = 0$$

Step 3: Check the value of the function at x=0

The function is defined as $f(0) = 0$.

Step 4: Compare LHL, RHL, and f(0)

We have $LHL = 0$, $RHL = 0$, and $f(0) = 0$. Since $LHL = RHL = f(0)$, the function is continuous at $x=0$.

Final Answer: The function is continuous at x=0.

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to demonstrate understanding of the definition of continuity and apply it to the given function. They must compute the left-hand limit, right-hand limit, and the function's value at the given point and then compare them.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concept of continuity, limits, and the application of the Squeeze Theorem. It's not just recalling facts but applying concepts to determine continuity.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the continuity of a function, a core concept in the syllabus.

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