How do you find the vertical and horizontal asymptotes
Find the vertical asymptotes of a rational function:
A rational function in lowest terms will have a vertical asymptote if is a real zero of the denominator . That is, if is a factor of the denominator of a rational function , in lowest terms, will have the vertical asymptote .
Ways to find the horizontal asymptote of a rational function
Consider the rational function in which the degree of the numerator is and the degree of the denominator is .
1. If (the degree of the numerator is less than the degree of the denominator), the line is a horizontal asymptote.
2. If (the degree of the numerator equals the degree of the denominator), the line is a horizontal asymptote. (that is, the horizontal asymptote equals the ratio of the leading coefficients.)
3. If (the degree of the numerator is one more than the degree of the denominator), then there is no horizontal asymptote. If in lowest terms is a linear
function, then we agree that has no horizontal asymptote.
4. If (the degree of the numerator is two or more greater than the degree of the denominator), there are no horizontal or oblique asymptotes.
For eg.
Vertical asymptotes of
is in lowest terms, and the zero of the denominator is 3. The lines is the vertical asymptote of the graph of .
Horizontal Asymptotes of
Since the degree of numerator is one more than the degree of the denominator, there is no horizontal asymptote.
Hence, the lines is the vertical asymptote of the graph of and there is no horizontal asymptote of .