How many vertical asymptotes can a rational function have?

1 Answer
Aug 20, 2015

Answer:

Assuming that you mean a function of the form #(p(x))/(q(x))# where #p(x)# and #q(x)# are polynomials, then the maximum number of vertical asymptotes is the degree of #q(x)#.

Explanation:

Suppose #f(x) = (p(x))/(q(x))# where #p(x)# and #q(x)# are polynomials.

If #q(x_1) = 0# and #p(x_1) != 0# then #f(x)# will have a vertical asymptote at #x = x_1#.

If #q(x_1) = 0# and #p(x_1) = 0# then #f(x)# will have a vertical asymptote at #x = x_1# if and only if the multiplicity of this zero of #q(x)# is greater than the multiplicity of this zero of #p(x)#.

So the maximum possible number of vertical asymptotes is the number of zeros of #q(x)#, which is at most the degree of #q(x)#.