UPSC NDA I 2023 Mathematics Solved Paper

Concept:The limit of a function involving absolute value depends on the side from which $x$ approaches the point. If the left-hand limit and right-hand limit are not equal, the limit does not exist.Explanation:We need to evaluate $\underset{x \to 5}{\lim} \frac{5 - x}{|x - 5|}$. Consider the left-hand limit as $x \to 5^{-}$ (values less than 5). For $x < 5$, $|x - 5| = 5 - x$. Then $\frac{5 - x}{|x - 5|} = \frac{5 - x}{5 - x} = 1$. So $\underset{x \to 5^{-}}{\lim} \frac{5 - x}{|x - 5|} = 1$. Now consider the right-hand limit as $x \to 5^{+}$ (values greater than 5). For $x > 5$, $|x - 5| = x - 5$. Then $\frac{5 - x}{|x - 5|} = \frac{5 - x}{x - 5} = - \frac{x - 5}{x - 5} = -1$. So $\underset{x \to 5^{+}}{\lim} \frac{5 - x}{|x - 5|} = -1$. Since the left-hand limit ($1$) is not equal to the right-hand limit ($-1$), the two-sided limit does not exist.Answer:The limit does not exist. (Option D)