Question

The set of all points of discontinuities of the $f\left(x\right)=\frac{\tan x\log x}{1-\cos 4x}$

• A. $\{n\frac{\pi }{2};nϵZ\}$
• B. $\{n\frac{\pi }{2};nϵQ\}$
• C. $(-\infty ,0)\cup \{n\frac{\pi }{2};nϵN\}$
• D. none of these

Answer 1

Answer: (A), (C)

$\{n\frac{\pi }{2};nϵZ\}$
$(-\infty ,0)\cup \{n\frac{\pi }{2};nϵN\}$

The points of discontinuity are when $1-\cos 4x=0$, or when $\tan x\to \pm \infty$, or when $x\le 0$ (since $\log x$ is not defined for such $x$).
These points are:
$1-\cos 4x=0\Rightarrow \cos 4x=1\Rightarrow 4x=2n\pi \Rightarrow x=\frac{n\pi }{2}\tan x\to \pm \infty \Rightarrow x=\frac{(2n+1)\pi }{2}$
Combining all points of discontinuity:
$x=\frac{n\pi }{2}$ and $x\le 0$.