AP Calculus BC / Unit 1: Limits and Continuity
1.13 Removing Discontinuities
Notes
A removable discontinuity can be "fixed" by defining (or redefining) $f(c)$ to equal the limit.
- Find the limit (often by factoring), then set $f(c)$ equal to it.
- Jump and infinite discontinuities can't be removed.
Practice
How should $f(4)$ be defined to make $\displaystyle f(x)=\frac{x^2-16}{x-4}$ continuous at 4?
The limit is $4+4=8$.
Which discontinuity can be removed by redefining one point?
Only a hole has a limit to fill in.
$\displaystyle f(x)=\frac{\sin x}{x}$ for $x\ne0$. Defining $f(0)=$ ___ makes $f$ continuous.
$\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1$.
Ready for more? Take the Unit 1 test.