Question

How do you find the Vertical, Horizontal, and Oblique Asymptote for `R(x)=(3x+5) \ (x-6)`?

Answer 1

This function has no asymptotes.

Explanation:

The given function is a polynomial and polynomials do not have asymptotes.

Answer 2

Was the question intended to be for the function `R(x) = (3x+5)/(x-6)`?

Explanation:

The quotient is already reduced. (There are no common factors onf the numerator and the denominator).
So we can find vertical asymptotes by solving

denominator = `0`.

The equation of the vertical asymptote is `x=6`.

For values of `x` far from `0` (positive or negative), we have

`R(x) = (cancel(x)(1+5/x))/(cancel(x)(1-6/x))`

For `x` far from `0` (`x` with large absolute value), both `5/x` and `6/x` are close to `0`, so `R(x)` is close to `3/1 = 3`.

The line `y=3` is a horizontal asymptote on both sides.

The graph of a rational function cannot have both horizontal and obliques asymptotes. The graph of this function does have horizontal asymptotes, so it cannot have oblique asymptotes.