Verify whether the function f defined by f(x)=casesx (1/x ),&x≠0\\0,&x=0cases is continuous at x=0 or not.

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2024: question 22(a)

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section B, question 22(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22(a) · 2 marks · Very short answerContinuity and Differentiability

Verify whether the function f defined by $f (x) = {x sin (\frac{1}{x}), 0, x \neq = 0 x = 0$ is continuous at $x = 0$ or not.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Continuity at 0 needs $lim_{x \to 0} f (x) = f (0)$ .

Hint 2

$sin (1/ x)$ always lies between $- 1$ and $1$ ; what happens when this is multiplied by something tending to 0?

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

For $x \neq = 0$ , $∣ sin (1/ x) ∣ \leq 1$ , so $0 \leq ∣ x sin (1/ x) ∣ \leq ∣ x ∣$ . As $x \to 0$ , $∣ x ∣ \to 0$ , hence $lim_{x \to 0} x sin \frac{1}{x} = 0$ (zero times a bounded quantity). Also $f (0) = 0$ .
1½ marks
Since $lim_{x \to 0} f (x) = 0 = f (0)$ , f is continuous at $x = 0$ .
½ mark

Answer: f is continuous at $x = 0$ .

Where the marks are

Shows $lim_{x \to 0} x sin \frac{1}{x} = 0$ using boundedness of sine and compares with $f (0) = 0$ (1½ marks)
Concludes f is continuous at $x = 0$ (½ mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 22(a)

Hints

Step-by-step solution

Revise the chapter

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