The end behavior of a rational function (what does as grows very large in magnitude) can be determined by the structure of the function's expression. This app demonstrate the three basic cases of horizontal or oblique (slant) asymptote based on the relative degrees of the numerator and denominator polynomials, and their leading coefficients.

The degrees and the leading coefficients of the numerator and denominator polynomials can be set with the four horizontal sliders. Each time a change is made to one of these, a new random function is generated. New functions can also be randomly generated using current slider settings by clicking the New f(x) button.
is graphed in solid red while the HA or OA is graphed in dashed green. The equation of the asymptote is also displayed. Note that the -value of the HA or the slope of the OA is determined solely by the leading coefficients, along with the relative degrees, of the numerator and denominator.
To see how the HA or OA approaches a single line as x grows in magnitude, adjust the Zoom slide to change the scale of the graph. The and scales are changed simultaneously to maintain the proper aspect ratio of the display.