AP Calculus AB · Unit 1: Limits & Continuity

1.4 Estimating Limits from Tables

Building the table

To estimate lim(x→c) f(x), evaluate f at x-values that get closer and closer to c from both sides. If the outputs close in on a single number, that number is your estimated limit.

Estimates, not proof

A table only estimates a limit — it never proves it exactly. If the left-side values head to one number and the right-side values head to another, the limit does not exist.

Key terms

Estimating from a table: Evaluate f at x-values approaching c from both sides.
Approaching from both sides: Needed to confirm the two-sided limit exists.
Convergence of values: If outputs close in on one number, that is the estimated limit.
Estimate vs. exact: Tables estimate a limit; they don't prove it.
Table shows DNE: Left and right values heading to different numbers.

Try an AP-style question

Data

x: 1.9, 1.99, 1.999, 2.001, 2.01, 2.1 → f(x): 3.61, 3.9601, 3.996, 4.004, 4.0401, 4.41

What is the best estimate of lim(x→2) f(x)?

3.996
4.41
4
2

Show the answer

C. Values approach 4 from both sides.

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