Understanding Limits: Graphs, Tables, and Notation

What a limit means

The limit of f(x) as x approaches a is the value f(x) gets close to when x gets close to a, from both sides. We write

lim (x→a) f(x) = L

The limit is about the approach, not the value at the point. f(a) can be different from L, or not exist at all, and the limit can still exist.

One-sided limits

• lim (x→a⁻) f(x) is the value approached from the left (x < a).
• lim (x→a⁺) f(x) is the value approached from the right (x > a).

The two-sided limit exists only when both one-sided limits exist and are equal.

Reading limits from a graph

1. Trace the curve toward x = a from the left; note the height it approaches.
2. Do the same from the right.
3. Compare. Ignore any open or filled dot exactly at x = a; that is f(a), not the limit.

Reading limits from a table

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	4.9	4.99	4.999	5.001	5.01	5.1

Values approach 5 from both sides, so the limit as x→2 appears to be 5.

When a limit does not exist

• Jump: the one-sided limits are different numbers.
• Unbounded: f(x) grows without bound near a (vertical asymptote).
• Oscillation: f(x) keeps jumping between values, as sin(1/x) does near 0.

AP exam tips

• Always state one-sided limits separately when a graph has a break.
• DNE (does not exist) must be justified: say which side gives which value.
• A hole in the graph does not prevent a limit from existing.

Quick summary

• A limit describes where f(x) is heading, not where it is.
• Two-sided limit exists if and only if the left and right limits agree.
• Jumps, vertical asymptotes and oscillation make a limit fail to exist.