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. For such huge values of x, only the biggest exponent will produce much change.

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 f(x) = (2x3 - 2)/(3x3 - 9).

The exponents in this case are the same in the numerator and denominator. See it? The exponent or power is the same (3). You can easily find the horizontal asymptote by simply dividing the coefficients. In this case, 2/3 is the horizontal asymptote of the above function. This is because when we approach x=infinity, the -2 and -9 are irrelevent. Then, the x^3 terms cancel out, and we are left with 2/3.

NOTE:

1) If there is bigger exponent in the denominator of the equation or function, the x-axis will yield the horizontal asymptote(s). LOOK: (2x - 3)/(x2 - 4). In this case, the exponent is bigger in the denominator. Thus, the x-axis or y = 0 will equal the horizontal asymptote for the above function.

2) If there is a bigger exponent in the numerator of a given equation or function, then there is NO horizontal asymptote whatsoever. For example, in the function y = (x3 - 27)/(2x2 - 4), there will be NO horizontal asymptote(s) because there is a BIGGER exponent in the numerator, which is 3. See it? (x cubed or x to the third power is BIGGER than x squared or x to the second power or x^3 > x^2).

REMEMBER: The word power = exponent = degree in mathematics.

Sample B: Find the horizontal asymptotes of f(x) = (2x - 1)(x + 3)/(x(x - 2))

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. See it?

Sample B, in standard form, looks like this:

f(x) = (2x2 + 5x - 3)/(x2 - 2x)

Next: Follow the steps above.

We drop everything except the biggest exponents of x found in the numerator and denominator. After doing so, the above function becomes: f(x) = 2x2/x2.

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 Mr. Feliz
(c) 2005