Question

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

$G\left(x\right)=\frac{x^3+1}{x^2-5x-14}$

Short answer

$y=x+5$ is the oblique asymptote for the given rational expression.

Step-by-step solution

Step 1. Given information

Given rational expression:

$G\left(x\right)=\frac{x^3+1}{x^2-5x-14}$

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,$n=3,m=2$

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

$$\begin{gathered} y=ax+bwillbetheobliqueasymptote \\ y=x+5 \end{gathered}$$

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