Option 9: Vertical Asymptotes (Limits that equal infinity)

See examples of this topic from AP Exams

If you find yourself unable to apply any of the previous 8 Limit Options; if you can’t seem to get rid of that piece of the equation that is giving you division by zero, then you are most likely looking at a vertical asymptote.

Definition: Vertical Asymptotes

Vertical asymptotes occur at an $x - value$ when the answer to a one-sided limit (left-hand limit or right-hand limit) at that $x - value$ is $\pm \infty$ .

LHL : $\lim_{x \to a^{-}} f ((x)) = + \infty$ or $\lim_{x \to a^{-}} f ((x)) = - \infty$

RHL : $\lim_{x \to a^{+}} f ((x)) = + \infty$ or $\lim_{x \to a^{+}} f ((x)) = - \infty$

Note:

– The answer to a vertical asymptote question is always the $x - value$ where it is occurring.

– The overall limit does not need to exist in order for there to be a vertical asymptote. The left-hand limit does not need to equal the right-hand limit.

– Only one of the limits (left or right) needs to equal $\pm \infty$ . They both do not need to equal $\pm \infty$ .

– Vertical asymptotes most often occur where you would get division by zero in your equation. They do not always occur there. Remember that if you are able to cancel out a part of an equation, then you have a removeable discontinuity (hole in the graph) not a vertical asymptote.

For Example : In this equation you would have a vertical asymptote at $x = - 9$ because can’t get rid of that part of the equation. However, you would have a removeable discontinuity at $x = 8$ because the $((x - 8))$ terms can be canceled.