Idea
An infinite limit describes what a function does near a point when the output grows without bound, such as becoming very large positive or very large negative. A vertical asymptote is a vertical line where the function values blow up to infinity or negative infinity near that x-value.
Method
- Identify the critical x-value where the expression may be undefined, often where a denominator is 0.
- Simplify first if possible. Factor, cancel common factors, or rewrite the expression to reveal the behavior more clearly.
- Check one-sided behavior. Look at values just to the left and right of the critical point.
- Determine the sign of the expression near that point. The sign tells you whether the limit is (+\infty) or (-\infty).
- State the conclusion clearly. If the function grows without bound on either side, the line x = a is a vertical asymptote.
How to check
- Substitute nearby values or use sign analysis on factors.
- If a factor cancels, the issue may be a hole instead of an asymptote, so do not assume vertical asymptotes too quickly.
- For the final response, write the limit in exact form, such as (\infty) or (-\infty), and simplify any algebraic expression before concluding.
Common mistake
Do not treat every undefined point as a vertical asymptote. First simplify and then examine the limit behavior.