Question

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

Answer 1

`\cot\frac{2x}3` is undefined wherever
`\sin\frac{2x}3=0`.

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


`\sin\frac{2x}3=0\implies\frac{2x}3=n\pi`

`\implies x=\frac{3n\pi}2`

where
`n` is any integer.

Answer 2

Explanation:

cot(x) can be written as

`cot(x) =(cosx)/(sinx)`

Here we have
`cot((2x)/(3)`

so It will be undefined whenever

`sin((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 ((2x)/(3) ) = 0`

`(2x)/(3) = 0 + \pi k` where k is any integer

`x= (3\pi )/(2) k`

It means f(x) is discontinuous at all the values of
`x = (3\pi )/(2)k`

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