Answer:
[tex]\displaystyle{x=\dfrac{\pi}{2}n}[/tex] for n is any integers
Step-by-step explanation:
To know the asymptotes, first, we must know values of x that we turn y-value into an undefined value.
We know that:
[tex]\displaystyle{\cot x = \dfrac{1}{\tan x}}[/tex]
Now we have to find value of x that turns the identity above into undefined value, and that is [tex]\displaystyle{x=n\pi}[/tex] where n is any integers. (This gives 1/0 for all x = nπ)
Therefore, a function [tex]\displaystyle{\cot x}[/tex] has asymptote lines at [tex]\displaystyle{x=n \pi}[/tex] for n is integers.
If we consider the given problem:
[tex]\displaystyle{y=\dfrac{1}{3}\cot 2x}[/tex]
We have to find values of x that turn y-value undefined. We know that [tex]\displaystyle{x=n\pi}[/tex] is asymptotes for [tex]\displaystyle{\cot x}[/tex]. Therefore, [tex]\displaystyle{2x=n\pi}[/tex] has to be asymptotes for [tex]\displaystyle{y=\dfrac{1}{3}\cot 2x}[/tex].
Hence, the asymptotes occur at [tex]\displaystyle{x=\dfrac{\pi}{2}n}[/tex] by solving the equation and for n is any integers.
