1. Tardigrade
  2. Question
  3. Mathematics
  4. Evaluate displaystyle limθ → (π/2) (sec θ-tan θ) is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Evaluate $\displaystyle \lim_{\theta \to \frac{\pi}{2}}$ $\left(sec\,\theta-tan\,\theta\right)$ is

Limits and Derivatives

Solution:

$\displaystyle \lim_{\theta \to \frac{\pi}{2}}$ $\left(sec\,\theta-tan\,\theta\right)$
$=\displaystyle \lim_{\theta \to \frac{\pi}{2}}$$\left(\frac{1}{cos\,\theta}-\frac{sin\,\theta}{cos\,\theta}\right)$
$=\displaystyle \lim_{\theta \to \frac{\pi}{2}}$$\frac{1-sin\,\theta}{cos\,\theta}$
$=\displaystyle \lim_{\theta \to \frac{\pi}{2}}$$\frac{cos^{2} \frac{\theta}{2}+sin^{2} \frac{\theta}{2}-2\cdot sin \frac{\theta}{2}\cdot cos \frac{\theta}{2}}{cos^{2}\, \frac{\theta}{2}-sin^{2}\, \frac{\theta}{2}}$
$=\displaystyle \lim_{\theta \to \frac{\pi}{2}}$$\left[\frac{\left(cos \frac{\theta}{2}-sin \frac{\theta}{2}\right)^{2}}{\left(cos \frac{\theta}{2}+sin \frac{\theta}{2}\right)\left(cos \frac{\theta}{2}-sin \frac{\theta}{2}\right)}\right]$
$=\displaystyle \lim_{\theta \to \frac{\pi}{2}}$$\frac{cos \frac{\theta }{2}-sin \frac{\theta }{2}}{cos \frac{\theta }{2}+sin \frac{\theta }{2}}=\frac{cos \frac{\pi}{4}-sin \frac{\pi}{4}}{cos \frac{\pi}{4}+sin \frac{\pi}{4}}$
$=\frac{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=0$