AP Precalculus / Unit 1: Polynomial and Rational Functions
1.10 Rational Functions and Holes
Notes
A hole happens at $x=a$ when $(x-a)$ cancels from numerator and denominator (same or higher multiplicity on top). The function is undefined there, but the limit exists.
- Find the hole's $y$-value by plugging $a$ into the simplified function.
Practice
Where is the hole in $\displaystyle \frac{x^2-x-2}{x-2}$?
Simplifies to $x+1$; at $x=2$, that's 3.
$\displaystyle \frac{(x+4)(x-1)}{(x+4)}$ has a hole at $x=$
The $(x+4)$ cancels.
A hole differs from a vertical asymptote because at a hole:
The limit exists at a hole.
Ready for more? Take the Unit 1 test.