Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.
$R\left(x\right)=\frac{6x^2+7x-5}{3x+5}$

Short answer

$y=2x-1$is the oblique asymptote of given rational expression.

Step-by-step solution

Step 1. Given information

Given rational expression:

$R\left(x\right)=\frac{6x^2+7x-5}{3x+5}$

Step 2. Write the concept

If n = m + 1 (the degree of the numerator is one more than the degree of the denominator), the

quotient obtained is of the form ax + b (a polynomial of degree 1), and the line y = ax + b is an oblique

asymptote.

Step 3. Calculation

Here,$n=2,m=1$

Thus, $n=m+1$and therefore,

$$\begin{gathered} y=ax+bwillbeobliqueasymptote. \\ y=2x-1 \end{gathered}$$

i.e. the quotient when we divide numerator with denominator.