1.2 Defining Limits and Using Limit Notation
- Syllabus
- 2020
- Topic
- 1.2
- Level
- —
Learning objectives
—LIM-1.A—Represent limits analytically using correct notation• LIM-1.A Represent limits analytically using correct notation.• LIM-1.A.1 Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is lim x→c f(x) = R.- Exclusion statement: The epsilon-delta definition of a limit is not assessed on the AP Calculus AB or BC Exam. However, teachers may include this topic in the course if time permits.• Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.—LIM-1.B—Interpret limits expressed in analytic notation• LIM-1.B Interpret limits expressed in analytic notation.• LIM-1.B.1 A limit can be expressed in multiple ways, including graphically, numerically, and analytically.• Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.
Writing a Limit Statement Correctly
A finite limit states the value that f(x) approaches when x is taken sufficiently close to a specified input, without requiring x to equal that input.
\lim_{x\to c}f(x)=R
x→c describes inputs approaching c; f(x) is the output being tracked; R is the real number approached by those outputs. Read the statement: “The limit of f(x) as x approaches c equals R.”
For f(x)=x2+1, values of x close to 2 produce values of f(x) close to 5, so limx→2(x2+1)=5. The notation reports nearby behavior, not an instruction to write only f(2)=5.
The input may approach c from values not equal to c. The AP Calculus AB/BC Exam does not assess the epsilon-delta definition, so correct notation and interpretation are required without that formal proof.
Interpreting Limits Across Representations
The statement limx→cf(x)=R has the same meaning in every representation: when inputs approach c, the corresponding outputs approach R.
| Representation | Evidence for the same limit |
|---|---|
| Analytical | The expression limx→cf(x)=R states the approaching input and output |
| Graphical | The graph's y-values approach R as x approaches c from both sides |
| Numerical | Table values of f(x) approach R for x-values increasingly close to c from below and above |
| Verbal | Outputs can be made arbitrarily close to R by taking inputs sufficiently close to c, with x=c |
If values in a table approach 7 as x approaches 4 from both sides, the analytic statement is limx→4f(x)=7. A graph representing the same fact approaches height 7 near x=4.
A limit describes nearby behavior. The value f(c) may equal R, may be different from R, or may be undefined; none of those facts alone changes the limit if nearby outputs still approach R.