Finding the Asymptotes to the Graphs of Rational Functions

Basic Concept of Asymptotes

An asymptote is a line that a curve approaches as it heads towards infinity.
There are three types of asymptotes: horizontal, vertical, and oblique.
Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function.
Horizontal asymptotes are horizontal lines which represent the limit of the function as it approaches positive or negative infinity.

To find the vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.
To find the horizontal asymptotes, compare the degrees of the numerator and the denominator.
- If the degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is the horizontal asymptote.
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator by one, there are no horizontal asymptotes. Instead, there is an oblique asymptote.

You can check for vertical asymptotes by substituting values close to the predicted asymptote into the function. As the values approach the asymptote, the function should evaluate to almost infinity or negative infinity.
To verify horizontal or oblique asymptotes, substitute large positive and negative x-values into the function. The function should converge to the predicted y-value.

Plot the rational function and asymptotes on the same graph.
The graph of the function will approach but never cross the asymptotes.
Graphs of rational functions may cross horizontal or oblique asymptotes in the finite x-region, but will approach the asymptotes at the extremes.
Ensure a clear understanding of how the graph behaves in the vicinity of the asymptotes, as this is part of the concept of limits in calculus.