Learning Objectives
Interpret a limit as the output approached near an input.
Distinguish a limit from a function value.
Use tables and one-sided behavior to decide whether a two-sided limit exists.
Prerequisites: function notation, substitution, and reading a coordinate graph. This section develops introductory limits and continuity shared by Calculus AB and BC.
1. A Limit Describes Nearby Behavior
The statement lim(x → a) f(x) = L means that the outputs of f can be made as close to L as desired by taking x sufficiently close to a, with x ≠ a. The value at a itself does not decide the limit. A graph can approach a height even if there is a hole at that point.
Read the notation as: the limit of f(x), as x approaches a, is L. The input a and the output L play different roles. When estimating from a graph, follow the curve toward x = a and read the height it approaches.
Example: A Hole in the Graph
Let f(x) = (x² − 4)/(x − 2), with x ≠ 2. Factor the numerator: x² − 4 = (x − 2)(x + 2).
For every permitted input, f(x) = x + 2. Therefore, as x approaches 2, the outputs approach 4.
Conclusion: lim(x → 2) f(x) = 4, although f(2) is undefined. Cancelling a factor does not fill the hole in the original function.
2. Read a Table from Both Sides
x | f(x) = x + 2, x ≠ 2 |
|---|---|
1.9 | 3.9 |
1.99 | 3.99 |
2 | Undefined |
2.01 | 4.01 |
2.1 | 4.1 |
The entries below 2 and above 2 both suggest an output of 4. A table is useful evidence, but finitely many values do not prove a limit. A function could behave differently between sampled inputs. Here the algebraic identity f(x) = x + 2 for x ≠ 2 justifies the conclusion.
3. Left and Right Must Agree
The left-hand limit lim(x → a⁻) f(x) uses inputs smaller than a. The right-hand limit lim(x → a⁺) f(x) uses inputs larger than a. A finite two-sided limit exists exactly when both one-sided limits exist and are equal.
Example: A Jump
Define g(x) = x + 1 for x < 2, and g(x) = 5 for x ≥ 2.
From the left, x + 1 approaches 3. From the right, the output remains 5.
Since 3 ≠ 5, lim(x → 2) g(x) does not exist. The actual value g(2) = 5 does not repair the disagreement.
4. Common Mistakes
Common Mistakes
Do not substitute the filled-dot value for the limit without checking nearby behavior.
Do not average different one-sided limits. Approaching 3 on one side and 5 on the other does not give a limit of 4.
An undefined function value does not automatically imply an undefined limit.
5. Practice
1. Find lim(x → 3) (2x − 1).
2. Let h(x) = x² for x ≠ 1, and h(1) = 7. Find h(1) and lim(x → 1) h(x).
3. A graph approaches height −2 from the left of x = 0 and height 2 from the right. Does its two-sided limit exist? Explain.
Worked Solutions
1. The limit is 5. As x approaches 3, multiplying by 2 and subtracting 1 makes the output approach 2(3) − 1 = 5.
2. h(1) = 7 by definition. Nearby inputs use x², so their outputs approach 1. Thus the limit is 1, even though the function value is 7.
3. No. The left-hand limit is −2 and the right-hand limit is 2. They are unequal.
6. Summary
- A limit concerns nearby outputs, not necessarily the value at the target input.
- Check both sides for an interior two-sided limit.
- Use tables to investigate, then justify with algebra or known function behavior.