Question

What happens with the horizontal asymptotes as you change the degree of the polynomial in the numerator and denominator in a rational function?

Answer 1

Well, a HA will happen where either the degr of num'r = degr of den'r, or degr of num'r < degr of den'r.

If degr of num'r < degr of den'r: then you get a HA at the line y=0, e.g, the x-axis is the HA. For large x, the den'r "takes over" the gets very large so the ratio approaches 0.

If degr of num'r = degr of den'r, then you get a HA at the line y=p/q, where p is the leading coefficient of the num'r and q is the leading coefficient of the den'r. That is, the leading terms take over both num'r and den'r, and the \(x^n\) parts "cancel" , so the function value settles in on the ratio of the leading coefficients.