Estimating Limit Values from Tables
Estimate a limit from both sides of a table and explain what finite samples cannot establish.
College Board Topic 1.4 · CED p. 36 (PDF p. 41) · CED effective Fall 2020; project course outline 2026–27.
Choose inputs approaching the target from both sides. Compare the trends in the outputs, keeping the target value in a separate row if it exists. A table supports an estimate; finitely many samples do not prove what happens at every nearby input.
| x | 1.9 | 1.99 | 2.01 | 2.1 |
|---|---|---|---|---|
| f(x) | 3.9 | 3.99 | 4.01 | 4.1 |
The data suggest lim(x→2)f(x)=4. If the rule is f(x)=(x²−4)/(x−2) for x≠2, algebra confirms f(x)=x+2 nearby. A calculator can supply the table, but rounding too early or entering x=2 causes trouble. Keep enough digits and avoid treating an undefined target output as evidence that the limit fails.
Let's work through it
- The left outputs 3.9 and 3.99 move toward 4. Reason: their inputs move toward 2 from below.
- The right outputs 4.01 and 4.1 are consistent with approaching 4 when read toward 2. Reason: approach order matters.
- Estimate the limit as 4; confirm with x+2 if the formula is supplied. Reason: a table gives evidence, while algebra gives a neighborhood argument.
Try it from memory
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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Why sample 1.999 and 2.001 rather than only 1.999?
See every recall question with its answer
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Retrieval 1
Why sample 1.999 and 2.001 rather than only 1.999?
Worked answer and reason
Both sides must be checked for a two-sided limit.
The other side might approach a different value.
Retrieval 2
Can four table entries prove a limit?
Worked answer and reason
No, though they can support a useful estimate.
Unsampled nearby inputs may behave differently.
Catch the tempting mistake
- A calculator error at x=c means the limit does not exist.
- The calculator may be attempting an undefined function value; nearby values may still approach a common number.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 1 problem 6 · No calculator · numerical
Explain and apply the mathematics
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Estimate nearby behavior from two displays
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For the teacher
- Use a graph/table/formula comparison to separate estimate from proof.
- Have students state the direction of approach and retain precision until the final estimate.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.