Question

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

$R\left(x\right)=\frac{3x}{x+4}$

Short answer

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

Step-by-step solution

Step 1. Given information

Given rational expression:

$R\left(x\right)=\frac{3x}{x+4}$

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,

$$\begin{gathered} R\left(x\right)=\frac{3x}{x+4} \\ takingxcommonweget \\ =\frac{3x}{x(1+{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$x$}\right.})} \\ =3 \end{gathered}$$