The vertical asymptote of the function is x=1.
The horizontal asymptote of the function is y=2.
What is an asymptote?
Asymptotes are straight lines that approach the graph of the given function. They appear to get closer and closer to the graph of the function but don’t touch the graph.
How to find an asymptote of a function?
Vertical Asymptotes:
If
[tex]\lim_{x\to b^-} f(x)=\pm\infty\\\lim_{x\to b^+} f(x)=\pm\infty[/tex]
then x=b is the vertical asymptote of the function f(x).
Horizontal Asymptotes:
If
[tex]\lim_{x\to \pm\infty} f(x)= a[/tex]
then y=a is the horizontal asymptote of the function f(x).
How to solve the problem?
Given function is f(x)=2x/(x-1)
Observe that,
[tex]\lim_{x\to 1^-} f(x)=-\infty\\\lim_{x\to 1^+} f(x)=\infty[/tex]
That means x=1 is a vertical asymptote of f(x).
Observe that,
[tex]\lim_{x\to\pm\infty}f(x)=2[/tex]
(How?)
By using the L'hospital rule
[tex]\lim_{x\to+\infty}\frac{2x}{x-1} =\frac{2}{1} =2\\\lim_{x\to-\infty}\frac{2x}{x-1} =\frac{2}{1} =2[/tex]
That is y=2 is a horizontal asymptote of f(x).
For more on asymptote visit- https://brainly.com/question/17767511?referrer=searchResults
#SPJ2
