Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x) = x / (3x-1)`?

Answer 1

vertical asymptote at `x=1/3`
horizontal asymptote at `y=1/3`

Explanation:

The denominator of f(x) cannot be zero as this would make f(x) 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: `3x-1=0rArrx=1/3" is the asymptote"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x.

`(x/x)/((3x)/x-1/x)=1/(3-1/x)`

as `xto+-oo,f(x)to1/(3-0)`

`rArry=1/3" is the asymptote"`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1) Hence there are no oblique asymptotes.
graph{(x)/(3x-1) [-10, 10, -5, 5]}