Question

q(x)= x 2 −6x+9 x 2 −8x+15 ​ q, left parenthesis, x, right parenthesis, equals, start fraction, x, squared, minus, 8, x, plus, 15, divided by, x, squared, minus, 6, x, plus, 9, end fraction Describe the behavior of the function qqq around its vertical asymptote at x=3x=3x, equals, 3.

1. According to the theory of rational functions, there are no vertical asymptotes at the rational function evaluated at x = 3.

### What is the behavior of a functions close to one its vertical asymptotes?

Herein we know that the rational function is q(x) = (x² – 6 · x + 9) / (x² – 8 · x + 15), there are vertical asymptotes for values of x such that the denominator becomes zero. First, we factor both numerator and denominator of the equation to see evitable and non-evitable discontinuities:
q(x) = (x² – 6 · x + 9) / (x² – 8 · x + 15)
q(x) = [(x – 3)²] / [(x – 3) · (x – 5)]
q(x) = (x – 3) / (x – 5)
There are one evitable discontinuity and one non-evitable discontinuity. According to the theory of rational functions, there are no vertical asymptotes at the rational function evaluated at x = 3.