1.2 Defining Limits & Limit Notation

What a limit is

A limit describes the value f(x) approaches as x gets arbitrarily close to a number c. It is written lim(x→c) f(x) = L, read 'the limit of f(x) as x approaches c is L.'

Approaching, not equaling

A limit is about what happens near c, not the value at c. The limit lim(x→c) f(x) can exist even if f(c) is undefined, because it only depends on nearby behavior.

Key terms

Limit

The value f(x) approaches as x gets arbitrarily close to c.

Limit notation

lim(x→c) f(x) = L: 'the limit of f as x approaches c is L.'

Approaches

The → symbol; x → c means x gets close to c without necessarily equaling it.

Limit vs. value at a point

A limit can exist even if f(c) is undefined or different.

Existence of a limit

Depends only on the behavior of f near c.

Try an AP-style question

What does lim(x→3) f(x) = 7 mean?

1. f(3) is equal to 7, whether or not f is continuous there
2. f(x) gets arbitrarily close to 3 as x gets close to 7
3. f(x) gets arbitrarily close to 7 as x gets close to 3 (x ≠ 3)
4. f(x) equals 7 for every x in some interval around 3