Finding Horizontal Asymptotes

A horizontal asymptote is a y-value which a function approaches but does not actually reach. Here is an example to make it obvious graphically:

To find horizontal asymptotes, we must write the function in standard form. Horizontal asymptotes take place as the graph of the function extends forever to the left or to the right. By this, I mean we are looking for very large positive or negative values of x.

To Find Horizontal Asymptotes:

1) Put equation or function in standard form.
2) Remove everything except the biggest exponents of x found in the numerator and denominator.

Sample A: Find the horizontal asymptotes of:

We need to figure out what this fraction approaches as x gets huge. To do that, we'll pick the "dominant" terms in the numberator and denominator. Dominant terms are those with the largest exponents. As x goes to infinity, the other terms are essentially meaningless.

The exponents in this case are the same in the numerator and denominator. See it? The dominant terms in each have an exponent of 3. Get rid of the other terms and you're left with:

In this case, 2/3 is the horizontal asymptote of the above function. You should actually express it as y=2/3. This value is the asymptote because when we approach x=infinity, the "dominant" terms will dwarf the rest and the function will always get closer and closer to y=2/3. Here's a graph of that function as a final proof that this is correct:

(Notice that there's also a vertical asymptote present in this function.)

If the exponent in the denominator of the function is larger than the exponent in the numerator, the horizontal asymptote will be y=0, which is the x-axis. As x approaches positive or negative infinity, that denominator will be much, much larger than the numerator (infinitely larger, in fact) and will make the overall fraction equal zero.

If there is a bigger exponent in the numerator of a given function, then there is NO horizontal asymptote. For example:

There will be NO horizontal asymptote(s) because there is a BIGGER exponent in the numerator, which is 3. See it? This will make the function increase forever instead of closely approaching an asymptote. The plot of this function is below:

Sample B: Find the horizontal asymptotes of:

In this sample, the function is in factored form. However, we must convert the function to standard form as indicated in the above steps before Sample A. That means we have to multiply it out, so that we can observe the dominant terms.

Sample B, in standard form, looks like this:

Next: Follow the steps from before. We drop everything except the biggest exponents of x found in the numerator and denominator. After doing so, the above function becomes:

Cancel x2 in the numerator and denominator and we are left with 2. Our horizontal asymptote for Sample B is the horizontal line y=2.

By Ted Wilcox and Mr. Feliz
(c) 2010