1.9 Connecting Representations of Limits

Three views agree

A limit can be studied from a graph, a table, or algebra — and all three should give the same value. Being fluent means translating between them.

Spotting DNE

A limit that does not exist shows up as a jump (different left/right behavior) in a graph, or as table values heading to two different numbers. Different representations describe the same underlying limit.

Key terms

Multiple representations

Graph, table, and algebra each reveal the same limit.

Agreement

All valid methods give the same limit value.

Jump = DNE

Different left/right behavior means the limit does not exist.

Numerical vs. algebraic

A table estimates; algebra confirms exactly.

Confirming a limit

Cross-check representations to verify a value.

Try an AP-style question

f(x) = x² for x < 1 and f(x) = 2x for x ≥ 1. What is lim(x→1) f(x)?

1. 1
2. 3/2
3. 2
4. Does not exist