AP Calculus BC / Unit 1: Limits and Continuity
1.9 Connecting Multiple Representations of Limits
Notes
A limit can be shown with a graph, a table, an equation or words, and all four should agree. Use one to check another.
- A table suggests a value; algebra confirms it.
- A graph shows holes, jumps and asymptotes that explain a table's behavior.
Practice
A table suggests $\displaystyle \lim_{x\to0}\frac{\sin(3x)}{x}\approx3$. Which confirms this algebraically?
Rewrite to use $\displaystyle \lim\frac{\sin u}{u}=1$.
A graph shows a hole at $x=2$ at height 7. A table near $x=2$ should show values close to:
Outputs near a hole approach the height of the hole.
Which representation can show a limit that the others can't?
They're different views of the same behavior.
Ready for more? Take the Unit 1 test.