NDA 1 2025 Maths, question 84

Topic: Differential calculus

Passage · Q83 to Q84

Let the function $f(x) = \sin\lbrack x\rbrack$ , where $\lbrack \cdot \rbrack$ is the greatest integer function and $g(x) = |x|$ .

Q84

What is $\lim_{x \rightarrow 0}\frac{f(x)}{g(x)}$ equal to?

(a) $- \sin 1$
(b) $\sin 1$
(c) 0
(d) Limit does not existCorrect answer

Solution.

Answer: (d) Limit does not exist

For $0 < x < 1$ , $\lbrack x\rbrack = 0$ , so the quotient is 0.
For $- 1 < x < 0$ , $\lbrack x\rbrack = - 1$ , so it is $- \sin 1/|x|$ , which tends to $- \infty$ .
The one-sided behaviours disagree; the two-sided limit does not exist.