AP Calculus AB / Unit 1: Limits and Continuity
1.7 Selecting Procedures for Determining Limits
Notes
Choose a method based on what substitution gives:
- A real number: that's the limit.
- $\displaystyle \frac{0}{0}$: factor, rationalize, simplify, or use a special trig limit.
- $\displaystyle \frac{\text{nonzero}}{0}$: the limit is infinite or doesn't exist; check one-sided signs.
- Piecewise functions: find each one-sided limit separately.
Practice
Substituting gives $\displaystyle \frac{3}{0}$. What does that tell you?
A nonzero number over something approaching 0 blows up.
For $f(x)=\begin{cases}x^2 & x<1\\ 2x & x\ge1\end{cases}$, what is $\displaystyle \lim_{x\to1}f(x)$?
Left limit is 1, right limit is 2.
Which limit is best found by direct substitution?
A polynomial is continuous everywhere.
Ready for more? Take the Unit 1 test.