Find all horizontal and vertical asymptotes if any If an ans

Find all horizontal and vertical asymptotes (if any). (If an answer does not exist, enter DNE. Enter your answers as a comma-separated list of equations.)

r(x) =

Solution

Let f(x) = (x³ +3x²)/(x² -25). Then f(x) is a rational function where the highest degree of the numerator is 3 and that of the denominator is 2 . Since the highest degree of the denominator is lower that that of the numerator, f(x) has no horizontal asymptote.

The vertical asymptotes , if any, are determined from the zeroes of the denominator. Here, the denominator is (x² -25) = (x+5)(x-5). Thus, the zeros of the denominator are 5 and -5.Then the vertical asymptotes of f(x) are x = 5 and x = -5.

Vertical Asymptotes

x = 5 and x = - 5

Horizontal Asymptotes

DNE