The end behavior of asymptotes of functions can be predicted using either polynomial long division or synthetic division. Finding the end behavior of asymptotes is valuable in circumstances where the degree of the numerator exceeds the degree of the denominator and neither term can be canceled out. Using the process of either polynomial long division or synthetic division produces the end product of an oblique asymptote, which is a type of linear function.
The final end linear function produced by these processes often creates a difference in degrees, where the degree of Q (x) exceeds the degree of R (x). The degrees of the two values create a linear function, expressed as y=ax+b. This linear function has the same limit as the original rational, and although approximations can be made for the value of the degrees, it is more useful for graphing purposes to find the specific oblique asymptote. The vertical lines in rational graphs are referred to as vertical asymptotes, and form when the denominator of the rational has no common factors with the numerator and the denominator equates to zero. The vertical asymptote is first identified, which then facilitates the task of finding the vertical asymptote. The vertical asymptote can be either positive or negative, which influences the pattern of the graph.