Defining Limits and Using Limit Notation
Distinguish a nearby limit from the value at the target, using one-sided and two-sided notation.
College Board Topic 1.2 · CED p. 34 (PDF p. 39) · CED effective Fall 2020; project course outline 2026–27.
A limit records what function outputs approach while inputs get close to a target. lim(x→c) f(x)=L says that f(x) can be made as close to L as desired by taking x sufficiently close to c, with x≠c. The value f(c) may agree, disagree, or be missing.
The symbol x→c⁻ means approach from smaller inputs; x→c⁺ means approach from larger inputs. A finite two-sided limit exists exactly when both one-sided limits exist and agree. Writing a two-sided limit does not silently choose the easier side.
Translate among representations: a graph shows the nearby height, a table shows nearby outputs, and a formula supports an exact argument. A verbal statement such as “temperature approaches 18°C as time approaches noon” becomes a limit with a specified function, input and output units. Formal epsilon–delta proofs are optional enrichment; this AP pathway uses the precise approaching-value idea without requiring those proofs.
Let's work through it
- Suppose f(x)=x+1 for x≠2 and f(2)=9. Read nearby values using x+1. Reason: the limit ignores the single target input.
- As x approaches 2 from either side, x+1 approaches 3. Reason: a linear function is continuous.
- Write lim(x→2)f(x)=3, while f(2)=9. Reason: the limit and the function value answer different questions.
Try it from memory
Pause & retrieve
Try it before you peek.
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
Translate “as t approaches 4, P(t) approaches 12.”
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Translate “as t approaches 4, P(t) approaches 12.”
Worked answer and reason
lim(t→4)P(t)=12.
The input target follows the arrow; the output target follows the equals sign.
Retrieval 2
If the left limit is 2 and the right limit is 7, is the two-sided limit 4.5?
Worked answer and reason
No. The two-sided limit does not exist.
One-sided limits must agree; averaging them is not a limit rule.
Catch the tempting mistake
- A limit exists only if f(c) exists.
- A missing value at c does not prevent nearby outputs from approaching a common value.
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
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 1 problem 2 · No calculator · verbal
- Unit 1 problem 3 · No calculator · analytic
For the teacher
- Ask students to say each symbol aloud before manipulating a limit.
- Distinguish a finite real limit from notation describing unbounded behavior.
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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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.
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One-sided limit
A one-sided limit uses inputs approaching a target from just one side. A left-hand limit uses smaller inputs, and a right-hand limit uses larger inputs. For example, a function that equals 0 left of 2 and 1 right of 2 has different one-sided limits at 2. The two-sided limit exists as a finite real number only when both one-sided limits exist and agree.