Differentiability

A function is said to be a continuous function at $x%3Da$ , if
$LHL%20%3D%20RHL%20%3D$ Value of function at $x%3Da%20.$
Since, $f(x)%3D%5Cbegin%7Bcases%7D%5Cfrac%7B%5Ctan%20x-%5Ccot%20x%7D%7Bx-%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%2C%20%26%20x%20%5Cneq%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5C%5C%20a%2C%20%26%20x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cend%7Bcases%7D$
$LHL%20%3D%5Cdisplaystyle%5Clim%20_%7Bh%20%5Crightarrow%200%7D%20f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D-h%5Cright)$
$%3D%5Cdisplaystyle%5Clim%20_%7Bh%20%5Crightarrow%200%7D%20%5Cfrac%7B%5Ctan%20%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D-h%5Cright)-%5Ccot%20%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D-h%5Cright)%7D%7B%5Cfrac%7B%5Cpi%7D%7B4%7D-h-%5Cfrac%7B%5Cpi%7D%7B4%7D%7D$
Applying L' Hospital's rule
$%3D%5Cdisplaystyle%5Clim%20_%7Bh%20%5Crightarrow%200%7D%20%5Cfrac%7B%5Csec%20%5E%7B2%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D-h%5Cright)%2B%5Ctext%7Bcosec%7D%5E%7B2%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D-h%5Cright)%7D%7B1%7D$
$%3D2%2B2%3D4%20.$
$RHL%20%3D%5Cdisplaystyle%5Clim%20_%7Bh%20%5Crightarrow%200%7D%20f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bh%5Cright)$
$%3D%5Cdisplaystyle%5Clim%20_%7Bh%20%5Crightarrow%200%7D%20%5Cfrac%7B%5Ctan%20%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bh%5Cright)-%5Ccot%20%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bh%5Cright)%7D%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bh-%5Cfrac%7B%5Cpi%7D%7B4%7D%7D$
$%3D%5Cdisplaystyle%5Clim%20_%7Bh%20%5Crightarrow%200%7D%20%5Cfrac%7B%5Csec%20%5E%7B2%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bh%5Cright)%2B%5Ctext%7Bcosec%7D%5E%7B2%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bh%5Cright)%7D%7B1%7D$
$%3D2%2B2%3D4$
$%5Cbecause$ Function is continuous at $x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D$ .
$%5Ctherefore%20f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)%3D%20RHL%20%3D%20LHL$
$%5Ctherefore%20a%3D4$
Alternate Method : $%5Cbecause%20f(x)$ is continuous function at $x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D$ .
$%5Ctherefore%20%5Cdisplaystyle%5Clim%20_%7Bx%20%5Crightarrow%20%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%20%5Cfrac%7B%5Ctan%20x-%5Ccot%20x%7D%7Bx-%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%3Da$
Applying L' Hospital's rule
$%5CRightarrow%20%5Cdisplaystyle%5Clim%20_%7Bx%20%5Crightarrow%20%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%20%5Cfrac%7B%5Csec%20%5E%7B2%7D%20x%2B%5Ctext%7Bcosec%7D%5E%7B2%7D%20x%7D%7B1%7D%3Da$
$%5CRightarrow%20a%3D2%2B2%3D4$

So, the correct option is (B): 4

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Differentiability

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs