UPSC NDA II 2024 Mathematics Paper

Concept:The limit of a function exists only if the left-hand limit and right-hand limit are equal.Explanation:We are evaluating $\lim_{x\to 3} \frac{|x-3|}{x-3}$.The absolute value $|x-3|$ behaves differently for $x<3$ and $x>3$.For $x \to 3$ from the left $(x<3)$, $|x-3| = -(x-3)$. Hence, the left-hand limit (LHL) is: $\displaystyle \lim_{x\to 3^-} \frac{-(x-3)}{x-3} = \lim_{x\to 3^-} (-1) = -1$.For $x \to 3$ from the right $(x>3)$, $|x-3| = x-3$. Hence, the right-hand limit (RHL) is: $\displaystyle \lim_{x\to 3^+} \frac{x-3}{x-3} = \lim_{x\to 3^+} (1) = 1$.Since the LHL = $-1$ and RHL = $1$ are not equal, the two-sided limit does not exist.Answer:The limit does not exist. Option D is correct.