AP Calculus BC / Unit 1: Limits and Continuity
1.10 Exploring Types of Discontinuities
Notes
- Removable: a hole; the limit exists but doesn't equal $f(c)$ (or $f(c)$ is undefined).
- Jump: the one-sided limits exist but differ.
- Infinite: a vertical asymptote; the function blows up.
Practice
$\displaystyle f(x)=\frac{x^2-4}{x-2}$ has what kind of discontinuity at $x=2$?
It simplifies to $x+2$ with a hole at $x=2$.
$\displaystyle g(x)=\frac{1}{x-3}$ has what kind of discontinuity at $x=3$?
The denominator goes to 0 while the numerator doesn't.
A step function like the greatest integer function has which type at each integer?
It jumps by 1 at each integer.
Ready for more? Take the Unit 1 test.