An asymptote is a horizontal/vertical oblique line whose distance from the graph of a function keeps decreasing and approaches zero, but never gets there.
In this wiki, we will see how to determine horizontal and vertical asymptotes in the specific case of rational functions. (Functions written as fractions where the numerator and denominator are both polynomials, like
For horizontal asymptotes in rational functions, the value of in a function is either very large or very small; this means that the terms with largest exponent in the numerator and denominator are the ones that matter. For example, with we only need to consider Since the terms now can cancel, we are left with which is in fact where the horizontal asymptote of the rational function is.
In a case like where there is only an term left in the denominator after the reduction process above, the horizontal asymptote is at 0.
In a case like where there is only an term left in the numerator after the reduction process above, there is no horizontal asymptote at all. (There may be an oblique or "slant" asymptote or something related.)
A vertical asymptote with a rational function occurs when there is division by zero. For example, with the denominator of is 0 when so the function has a vertical asymptote at
Find the vertical asymptote of the graph of the function
The denominator when Thus the line is the vertical asymptote of the given function.
Find the vertical asymptotes of the graph of the function
when that is, when and Thus this is where the vertical asymptotes are.