Describe the vertical asymptotes) and holes) for the graph of y = (x - 3)(x - 1)/(x - 1)(x - 5)

Solution:

A line x = a is a vertical asymptote of the graph of a function y = f(x) if either

\(lim_{x \rightarrow a+}f(x)) = \pm \infty\)

Or

\(lim_{x \rightarrow a-}f(x)) = \pm \infty\)

The vertical asymptotes of the given function x = 5.

Since the factor (x - 1) is both in the numerator and the denominator we define x =1 as a hole. At x = 1 the value of the function is not defined.

Describe the vertical asymptotes) and holes) for the graph of y = (x - 3)(x - 1)/(x - 1)(x - 5)

Summary:

The vertical asymptotes) and holes) for the graph of y = (x - 3)(x - 1)/(x - 1)(x - 5) are x = 5 and x = 1 respectively.