UPSC NDA II 2024 Mathematics Paper

Concept:The limit of a trigonometric expression that reduces to an indeterminate form can be evaluated using L'Hospital's rule after rewriting it as a single fraction.Explanation:We need to find $\displaystyle \lim_{\theta \to \frac{\pi}{2}} (\sec\theta - \tan\theta)$.Rewrite the expression: $\sec\theta - \tan\theta = \frac{1}{\cos\theta} - \frac{\sin\theta}{\cos\theta} = \frac{1-\sin\theta}{\cos\theta}$.At $\theta = \frac{\pi}{2}$, both numerator and denominator approach $0$, giving the indeterminate form $\frac{0}{0}$.Apply L'Hospital's rule: differentiate numerator and denominator separately.Derivative of $(1-\sin\theta)$ is $-\cos\theta$, and derivative of $\cos\theta$ is $-\sin\theta$.Thus the limit becomes $\displaystyle \lim_{\theta \to \frac{\pi}{2}} \frac{-\cos\theta}{-\sin\theta} = \lim_{\theta \to \frac{\pi}{2}} \frac{\cos\theta}{\sin\theta}$.Now substitute $\theta = \frac{\pi}{2}$: $\cos\frac{\pi}{2}=0$, $\sin\frac{\pi}{2}=1$, giving $0$.Hence the value of the limit is $0$.Answer:$0$ (Option B).