Question

The value of $\underset{\theta \to \frac{\pi }{2}}{lim}(\sec \theta -\tan \theta )$ equals

• A. $0$
• B. $1$
• C. $2$
• D. $\infty$

Answer 1

The correct option is A $0$

$\Rightarrow$ $\underset{\theta \to \frac{\pi }{2}}{lim}$ $(\sec \theta -\tan \theta )$

$\Rightarrow$ $\underset{\theta \to \frac{\pi }{2}}{lim}$ $(\frac{1}{\cos \theta }-\frac{\sin \theta }{\cos \theta })$

$\Rightarrow$ $\underset{\theta \to \frac{\pi }{2}}{lim}$ $\left(\frac{1-\sin \theta }{\cos \theta }\right)$

Applying L'Hospital rule, we get

$\Rightarrow$ $\underset{\theta \to \frac{\pi }{2}}{lim}$ $\left(\frac{\frac{d}{d\theta }(1-\sin \theta )}{\frac{d}{d\theta }\cos \theta }\right)$

$\Rightarrow$ $\underset{\theta \to \frac{\pi }{2}}{lim}$ $\frac{\cos \theta }{\sin \theta }$

$\Rightarrow$ $\underset{\theta \to \frac{\pi }{2}}{lim}$ $\left(\cot \theta \right)$

$\Rightarrow$ $\cot \frac{\pi }{2}$

$\Rightarrow$ $0$