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

Sep 15, 2015

This function has no asymptotes.

#### Explanation:

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

Sep 15, 2015

Was the question intended to be for the function $R \left(x\right) = \frac{3 x + 5}{x - 6}$?

#### 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

$R \left(x\right) = \frac{\cancel{x} \left(1 + \frac{5}{x}\right)}{\cancel{x} \left(1 - \frac{6}{x}\right)}$

For $x$ far from $0$ ($x$ with large absolute value), both $\frac{5}{x}$ and $\frac{6}{x}$ are close to $0$, so $R \left(x\right)$ is close to $\frac{3}{1} = 3$.

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.