AP Calculus AB · Unit 1: Limits & Continuity

1.3 Estimating Limits from Graphs

One-sided limits

From a graph, see what y-value the curve approaches. The left-hand limit lim(x→c⁻) uses values with x < c; the right-hand limit lim(x→c⁺) uses x > c. The two-sided limit exists only if the left and right limits are equal.

Reading holes and dots

An open circle (hole) means f is undefined there, but the limit can still exist. A filled dot at a different height sets f(c) but does not change the limit — the limit follows the curve, not the isolated point.

Key terms

One-sided limit: The value f approaches from just one side of c.
Left-hand limit: lim(x→c⁻): approaching c through x-values less than c.
Right-hand limit: lim(x→c⁺): approaching c through x-values greater than c.
Existence rule: The two-sided limit exists only if left limit = right limit.
Reading a hole: An open circle means f(c) is undefined; the limit can still exist.

Try an AP-style question

Graph

The graph of f has an open circle at (2, 3) and a filled point at (2, 5). The curve approaches the open circle from both the left and the right.

What is lim(x→2) f(x)?

4
Does not exist
3
5

Show the answer

C. The limit is the y-value the curve approaches (3); the filled point is f(2) = 5, which doesn't affect the limit.

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