1.2 Defining Limits and Using Limit Notation

What a limit means

The limit of f(x) as x approaches c describes the value f(x) gets close to as x gets arbitrarily close to c — from both sides. Notation: lim(x→c) f(x) = L. This value does NOT depend on f(c) itself; the function need not even be defined at c.

Two-sided vs one-sided limits

A two-sided limit exists only if the left-hand limit (x→c⁻) and right-hand limit (x→c⁺) both exist and are equal. If they differ, the overall limit does not exist (DNE). This idea is key for piecewise functions and jump discontinuities.

Estimating limits from graphs and tables

You can estimate limits by tracing a graph as x approaches c from each side, or by evaluating a table of function values getting closer to c. Look for the y-values to stabilize near a single number — that number is the limit estimate.

Key terms

Limit notation

lim(x→c) f(x) = L means f(x) approaches L as x approaches c.

One-sided limit

The value f(x) approaches as x approaches c from only the left (x→c⁻) or only the right (x→c⁺).

Condition for two-sided limit to exist

The left-hand and right-hand limits must exist and be equal.

Does a limit depend on f(c)?

No — a limit only depends on the values of f near c, not on f(c) itself.

DNE (Does Not Exist)

Occurs when the left-hand and right-hand limits are unequal, or the function oscillates/grows unbounded near c.

Try an AP-style question

What does lim(x→3) f(x) = 7 mean?

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