AP Calculus AB/BC · Unit 1 · Topic 1.2
Defining Limits and Using Limit Notation
Interpret two-sided and one-sided limit notation without confusing a limit with a function value.
1. Topic Focus
Build the language of limits, connect numerical, graphical, and algebraic representations, and use continuity theorems with verified hypotheses.
This topic: Interpret two-sided and one-sided limit notation without confusing a limit with a function value.
2. Key Relationship
Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.
3. Visual Connection
4. Worked Example
If both one-sided limits equal 5 while f(a)=1, the two-sided limit is still 5.
Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.
5. Concept Development
Read the entire statement as one claim
The notation
means that the outputs \(f(x)\) can be made as close as desired to \(L\) by taking inputs \(x\) sufficiently close to \(a\), from both sides, while considering nearby inputs rather than the input \(a\) itself.
| Symbol | Role | How to read it |
|---|---|---|
| \(\lim\) | Limit operator | Describes approaching behavior |
| \(x\to a\) | Input process | \(x\) approaches \(a\) from both sides |
| \(f(x)\) | Output expression | The quantity whose behavior is tracked |
| \(L\) | Approached value | The single real number approached by the outputs |
The arrow is a process, not an equation: \(x\to a\) does not mean \(x=a\). Also, the outputs are allowed to equal \(L\) near \(a\); they simply do not have to equal it.
Nearby behavior and the deleted neighborhood
A finite limit depends on values in a punctured, or deleted, neighborhood around \(a\): inputs close to \(a\), with \(x=a\) omitted. Therefore the limit statement alone makes no claim about the assigned value \(f(a)\).
The function may be undefined at \(a\), may have a different value there, or may equal its limit there. All three cases can share the same nearby behavior.
| Situation at \(x=a\) | Can \(\lim\limits_{x\to a}f(x)=L\) still hold? | Reason |
|---|---|---|
| \(f(a)=L\) | Yes | The point agrees with the nearby trend. |
| \(f(a)\ne L\) | Yes | One isolated value does not change nearby outputs. |
| \(f(a)\) is undefined | Yes | The function only needs values arbitrarily close to \(a\). |
One-sided limit notation
A superscript on the target tells which side supplies the inputs:
- \(x\to a^-\): use inputs \(x<a\), approaching from the left.
- \(x\to a^+\): use inputs \(x>a\), approaching from the right.
- The minus and plus signs describe the side of the input, not the sign of \(a\), \(f(x)\), or the limit.
- For example, \(x\to(-3)^+\) means values greater than \(-3\), such as \(-2.9\) and \(-2.99\).
When a two-sided limit exists
A two-sided finite limit exists exactly when the two one-sided limits both exist as finite values and agree:
If the sides approach different numbers, report both one-sided limits and then state that the two-sided limit does not exist. Do not average the two values.
Finite limits, infinite behavior, and DNE
The equation \(\lim\limits_{x\to a}f(x)=L\) uses a finite real number \(L\). Other notation communicates different behavior:
- \(\lim\limits_{x\to a^+}f(x)=+\infty\) means outputs increase without bound from the right; infinity is not a real output value.
- \(\lim\limits_{x\to a^-}f(x)=-\infty\) means outputs decrease without bound from the left.
- Write DNE when the two sides disagree, the function oscillates without settling, or no single limiting behavior can be assigned.
For \(r(x)=1/(x-4)\), the left side tends to \(-\infty\) and the right side tends to \(+\infty\), so the two-sided limit at 4 is DNE even though both one-sided behaviors can be described precisely.
Translate among representations
| Given information | Limit statement |
|---|---|
| A graph approaches height 6 from both sides of \(x=2\) | \(\lim\limits_{x\to2}f(x)=6\) |
| Table values for \(x<5\) approach \(-1\) | \(\lim\limits_{x\to5^-}g(x)=-1\) |
| Outputs grow without bound as \(x\) approaches 0 from the right | \(\lim\limits_{x\to0^+}h(x)=+\infty\) |
| Left outputs approach 3 and right outputs approach 8 | \(\lim\limits_{x\to a}p(x)\text{ DNE}\) |
Notation and communication traps
- Do not write \(\lim f(x)\) without stating what the input approaches.
- Do not replace a limit with \(f(a)\) unless continuity or another valid reason is known.
- Do not write \(x=a^-\); the superscript belongs in the approach notation \(x\to a^-\).
- Do not say “the limit is undefined” when the more precise conclusion is that the limit does not exist.
- When one-sided limits differ, include their values as justification for DNE.
AP scope note: an intuitive understanding and correct interpretation of limit notation are central here; a formal epsilon-delta proof is not required for this topic.
6. Detailed Worked Example and Error Check
Example 1: A limit can differ from the function value. Define
Approach from the left. Inputs less than 2 use \(x+3\), whose outputs approach 5:
Approach from the right. Inputs greater than 2 use \(7-x\), whose outputs also approach 5:
Because the one-sided limits agree,
However, the middle piece gives \(f(2)=9\). The complete conclusion is: the limit exists and equals 5, but the function value is 9.
Example 2: Unequal sides force DNE. Now define
Since \(5\ne2\),
The assigned value \(g(2)=9\) cannot repair the disagreement between the two nearby branches.
Example 3: Read a negative target correctly.
This says that as \(x\) approaches \(-3\) through values greater than \(-3\), the outputs approach 4. It does not say that the outputs are positive-side values, and it gives no information about \(k(-3)\) or the left-hand limit.
7. AP Reasoning Routine
Read one-sided behavior first, choose a matching limit procedure, and justify conclusions with definitions or theorem conditions.
- Identify the representation and requested quantity.
- State the rule or theorem and verify its conditions.
- Keep exact values until the final requested approximation.
- Interpret sign, units, interval, and context.
Let \(H(x)=2x+1\) for \(x<3\), \(H(3)=-5\), and \(H(x)=x^2-2\) for \(x>3\).
(a) Write and evaluate both one-sided limits at 3.
(b) State the two-sided limit and justify it.
(c) State \(H(3)\) and explain why it does not change the limit.
(d) Translate \(\lim\limits_{x\to3^-}H(x)=7\) into a complete sentence.
(e) If the right-hand rule were changed to \(x+5\), state the new right-hand and two-sided limits.
Check the solution
The left-hand limit is \(\lim\limits_{x\to3^-}H(x)=2(3)+1=7\), and the right-hand limit is \(\lim\limits_{x\to3^+}H(x)=3^2-2=7\). Since they agree, \(\lim\limits_{x\to3}H(x)=7\). The assigned value is \(H(3)=-5\), which does not affect nearby behavior. In words, as \(x\) approaches 3 through values less than 3, \(H(x)\) approaches 7. With the right-hand rule \(x+5\), the right-hand limit would be 8; because \(7\ne8\), the two-sided limit would be DNE.