NDA 1 2025 Maths, question 83

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 $g(x) = |x|$ .

Q83

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

(a) -1
(b) 0Correct answer
(c) 1
(d) Limit does not exist

Solution.

Answer: (b) 0

Since $\left| \sin\lbrack x\rbrack \right| \leq 1$ , we have $\left| f(x)g(x) \right| \leq |x|$ .
The right side tends to zero, so the squeeze theorem gives the limit 0.