Calculus BC · Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10
Connecting infinite limits and vertical asymptotes — AP Calculus BC
AP Calculus BC · Unit 1: Limits and Continuity · 14 min read
1. Infinite Limits: Definition and One-Sided Behavior★★☆☆☆⏱ 4 min
Infinite limits are almost always evaluated one-sided at a point, because the sign of the result depends which side of $a$ you approach from. For rational functions, if the numerator approaches a non-zero constant and the denominator approaches 0, the result is an infinite limit, with sign determined by the sign of the numerator and denominator near $a$.
Exam tip: Always evaluate one-sided limits separately when checking for infinite behavior. The AP exam frequently tests whether you recognize that a two-sided limit may not exist (because the two sides go to opposite infinities) but the one-sided behavior still creates a vertical asymptote.
2. Core Connection: Vertical Asymptotes from Infinite Limits★★☆☆☆⏱ 4 min
For rational functions, this gives a step-by-step rule to locate vertical asymptotes: (1) Factor numerator and denominator completely. (2) Cancel common factors to simplify; common factors create removable discontinuities (holes), not asymptotes. (3) Any $x=a$ that makes the simplified denominator zero is a vertical asymptote.
Exam tip: AP exam questions almost always include a common factor in rational function vertical asymptote problems to test if you confuse holes with vertical asymptotes. Always simplify first before identifying asymptotes.
3. Vertical Asymptotes of Non-Rational Functions★★★☆☆⏱ 4 min
Rational functions are not the only functions with vertical asymptotes. For any function, find points where the function is undefined, then check if at least one one-sided limit at that point is infinite. Two common cases tested on the AP exam are:
**Logarithmic functions**: For $f(x) = \ln(g(x))$, $f$ is undefined when $g(x) \leq 0$. Vertical asymptotes occur at $x=a$ where $g(x)$ approaches 0 from the positive side (within the domain of $f$).
**Trigonometric functions**: Reciprocal trigonometric functions like $\tan x$, $\cot x$, $\sec x$, and $\csc x$ have vertical asymptotes where their denominators are zero, since the numerator is non-zero at these points.
Exam tip: For logarithmic functions, only check boundaries of the domain where the argument approaches 0 from the positive side. Points where the argument approaches 0 from the negative side are outside the domain, so no asymptote exists there.
4. AP-Style Worked Practice★★★☆☆⏱ 6 min
Common Pitfalls
Why: Students confuse undefined points with asymptotes, forgetting to check for common factors that create removable discontinuities.
Why: Students incorrectly believe both one-sided limits must go to the same infinity for an asymptote to exist.
Why: Students confuse 'the limit does not exist as a finite number' with 'no infinite behavior that creates an asymptote'.
Why: Students memorize 'denominator zero means vertical asymptote' without checking the limit.
Why: Students forget to check what input makes the argument of the logarithm zero.