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:
**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风格例题练习★★★☆☆⏱ 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.