Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.

$R\left(x\right)=\frac{3x+5}{x-6}$

Short answer

$y=3$ is the horizontal asymptote for the given rational expression.

Step-by-step solution

Step 1. Given information

Given rational expression:

$R\left(x\right)=\frac{3x+5}{x-6}$

Step 2. Write the concept

If n = m (the degree of the numerator equals the degree of the denominator), the quotient obtained will be

the number $\frac{a_n}{b_m}$, and the line $y=\frac{a_n}{b_m}$is a horizontal asymptote.

Step 3. Calculation

Here, $n=1,m=1$

Now,

$$\begin{gathered} R\left(x\right)=\frac{3x+5}{x-6} \\ Takingxcommonweget \\ =li{m}_{x\to \infty }\frac{x(3+{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$x$}\right.})}{x(1-{\displaystyle \raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$x)$}\right.}} \\ =3 \end{gathered}$$