Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x)= 1/(x-2)`?

Answer 1

vertical asymptote x = 2
horizontal asymptote y = 0

Explanation:

The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: x - 2 = 0 → x = 2 is the asymptote

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" (a constant)"`

divide terms on numerator/denominator by x

`(1/x)/(x/x-2/x)=(1/x)/(1-2/x)`

as `xto+-oo,f(x)to0/(1-0)`

`rArry=0" is the asymptote"`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 0 , denominator-degree 1 ) Hence there are no oblique asymptotes.
graph{(1)/(x-2) [-10, 10, -5, 5]}