Rational functions have vertical asymptotes where the denominator equals zero, but the numerator does not.
This happens because division by zero is undefined, causing the function's value to approach infinity or negative infinity as it nears the asymptote.
If both the numerator and denominator have a common factor, cancel it first; otherwise, you might mistake a hole for an asymptote.
The key mechanism is comparing the multiplicities of the zero in the numerator and denominator.
If a zero has a higher multiplicity in the denominator, that's your vertical asymptote.
Misunderstanding arises when you conflate zeros and asymptotes.
A zero in the numerator just means the function crosses the x-axis there; it doesn't affect vertical asymptotes unless that zero's multiplicity matches the denominator's.
Watch out: near the asymptote, the function's behavior can flip from positive to negative rapidly, depending on the sign of the approaching values.
This concept is foundational for graphing rational functions and predicting their behavior.