Vertical Asymptotes

Asymptotes are lines that the graph of a function approaches. A vertical line in the x-y plane is a line that goes to plus and minus infinity. So vertical asymptotes indicate that the values of a function approach infinity (or minus infinity) as x approaches a finite value.

Example notes: The function f(x) = 1/(x – 1) has a vertical asymptote at x = 1. But the function g(x) = (x^{2} – 1)/(x – 1) does not have a vertical asymptote at x = 1. As long as x is not equal to 1, the function g(x) is equivalent to x + 1, which has no vertical asymptote. (Factor and simplify g(x) to make sure that this is true.)

Which functions below have vertical asymptotes? If there are any vertical asymptotes, what are their equations?

1. f(x) = 3 / (x – 2)

2. f(x) = (x^{3} + x)/x

3. f(x) = x / (x^{2} + 5)

4. f(x) = (2x^{3} – 32) / (x^{2} – 6x + 5)

5. f(x) = 1 – 1/x + 2/(x – 1)