Question

$\underset{x\to \mathrm{\pi }/4}{\lim }\frac{{\sec }^{2}x-2}{\tan x-1}\mathrm{is}$

​(a) 3
(b) 1
(c) 2
(d) $\sqrt{2}$

Answer 1

$$\begin{gathered} \begin{array}{c}\lim \\ x\to \frac{\mathrm{\pi }}{4}\end{array}\frac{{\sec }^{2}x-1-1}{\tan x-1} \\ \left(\mathrm{since}{\sec }^{2}x-1={\tan }^{2}x\right) \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to \frac{\mathrm{\pi }}{4}\end{array}\frac{{\tan }^{2}x-1}{\tan x-1} \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to \frac{\mathrm{\pi }}{4}\end{array}\frac{\left(\tan x-1\right)\left(\tan x+1\right)}{\tan x-1} \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to \frac{\mathrm{\pi }}{4}\end{array}\tan x+1=1+1=2 \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to \frac{\mathrm{\pi }}{4}\end{array}\frac{{\sec }^{2}x-2}{\tan x-1}=2 \end{gathered}$$

Hence, the correct answer is option C.