Question

List the discontinuities for the function f(x) = cot(2x over 3)

Answer 1

$\cot\dfrac{2x}3$ is undefined wherever $\sin\dfrac{2x}3=0$.

As $\sin x=0$ whenever $x=n\pi$ for any integer $n$, we have 

$\sin\dfrac{2x}3=0\implies\dfrac{2x}3=n\pi$
$\implies x=\dfrac{3n\pi}2$

where $n$ is any integer.

Answer 2

Answer:

Step-by-step explanation:

cot(x) can be written as

$cot(x) =\frac{cosx}{sinx}$

Here we have $cot(\frac{2x}{3}$

so It will be undefined whenever

$sin(\frac{2x}{3}) = 0$

As we cannot have a 0 in the denominator .

so to point all the discontinuties we need to identify the x values where

$sin (\frac{2x}{3} ) = 0$

$\frac{2x}{3} = 0 + \pi k$ where k is any integer

$x= \frac{3\pi }{2} k$

It means f(x) is discontinuous at all the values of $x = \frac{3\pi }{2}k$

for k = 0 , 1 ,2.....