How to get asymptotes of Rational Function


There are 2 kinds of asymptotes, vertical and horizontal ones.

Vertical Asymptote

Mathematically any denominator cannot be zero. If you know the x value which makes the denominator zero, that is an asymptote.

Example 1


The denominator is \(2x+6\)

Set \(2x+6=0\)

Solve for x


This is the vertical asymptote of this function.

Horizontal Asymptote

When \(x→∞\), the graph would approach to the horizontal asymptote. So we should deal with a limit.

Let’s think of the situation that the value of the x approaches to ∞.

Example 2

y=\frac{2x-1}{2x+6}-4 $$

But we have to arrange this fraction to deal with ∞

Divide by the x which has the largest exponent in the fraction. In this case x is good.

$$y=\frac{2x-1}{2x+6} \frac{(\frac{1}{x})}{(\frac{1}{x})} -4 $$


Take the limit


Since \(\lim_{x\to ∞} \frac{1}{x}=0\)

The equation is going to be



This calculation means the graph approaches to the line \(y=-3\), but never touch it.