Unit 1Behavior Near a Point

Concept 4.110Built for May 2027 AP Exam

Behavior Near a Point

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Near-point behavior describes what the outputs do as the input enters a punctured neighborhood of $a$ , the interval around $a$ with the single input $x = a$ removed. The point value $f(a)$ never enters that description. Whether $f(a)$ is defined, undefined, or printed far from the curve, the limit asks only which single finite height $L$ the outputs crowd toward from below and from above. A missing value at $a$ therefore costs nothing, while two arms ending at different heights cost everything. On exam day the split appears in piecewise definitions, data tables, and coordinate graphs, and the planted value at $a$ is usually one of the choices.

On the AP Exam

Exam writers build these items so that the value at the point and the behavior near the point disagree. On Section I you will meet graphs with an open circle and a detached dot, tables with an out-of-pattern entry in the center column, and rational functions with a removable discontinuity. On Section II, any continuity justification starts by establishing the two-sided limit, so reading the dot first costs the justification point immediately.

What you’ll see: Piecewise definitions that give $x = a$ its own rule, graphs pairing an open circle with a detached solid dot, and tables with one entry that breaks the pattern.
What you’ll do: Remove $x = a$ from the workspace, trace the left and right approaches separately, and decide whether both sides settle on one finite number.
Memory hook: $f(a)$ is the point; limit is the path. Ignore the point for the limit.
Use it on the exam: For MCQs: Before reading the choices, trace the curve from the left and from the right into the vertical line $x = a$ . The height where the two approaches meet is the limit. Any choice equal to $f(a)$ is the planted distractor, and so is the average of the two approaches when they disagree. For FRQs: Never justify a limit by saying the function is defined at $a$ . Write the two one-sided equations with their symbols, $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$ , then state that the values agree.

Concept Review

Core Term Definitions

Punctured Neighborhood of $a$ : An open interval around $a$ with $a$ itself removed, written $(a - \delta, a) \cup (a, a + \delta)$ or $0 < |x - a| < \delta$ . Limits operate here and nowhere else.
Behavior at the Point, $f(a)$ : The single output attached to the input $x = a$ . It is one number about one input, and it carries no information about the surrounding inputs.
Behavior Near the Point, $\lim_{x \to a} f(x)$ : The single real number $L$ that the outputs approach as the input runs through the punctured neighborhood from both sides.

Figure 1 — One graph, two independent readings.

Caption. Both arms close in on the open circle at height $3$ , so $\lim_{x \to 2} f(x) = 3$ . The filled dot reports $f(2) = 1$ . Erasing the dot would not move the arms, and raising the dot to $3$ would not change the limit.

Graph with an open circle at height 3 and a filled dot at height 1 at x equals 2

Step-by-Step Inspection Process

Run these four steps in order and the two facts stop mixing.

Fix the target input. Read off $a$ from $x \to a$ , then mark the two sides of the punctured neighborhood, $x < a$ and $x > a$ .
Remove the point. Cross out the row for $x = a$ in a table, cover the plotted dot at $(a, f(a))$ on a graph, or set aside the branch written for $x = a$ in a piecewise definition.
Trace each side alone. On the left, record where the outputs head as $x \to a^-$ . On the right, record where they head as $x \to a^+$ . Do not consult $f(a)$ while doing this.
Compare the two values. Equal finite values give $\lim_{x \to a} f(x) = L$ . Unequal values, an unbounded side, or oscillation give a limit that does not exist.

At Point vs. Near Point Comparison Table

Diagnostic Feature	At the Point: $f(a)$	Near the Point: $\lim_{x \to a} f(x)$
Input condition	Evaluates at $x = a$ exactly.	Evaluates for $x \neq a$ inside $0 < \lvert x - a \rvert < \delta$ .
Mathematical meaning	The plotted coordinate $(a, f(a))$ .	The height $L$ that the path approaches.
Dependence on other inputs	None. It is independent of every surrounding value.	Complete. It is determined only by values arbitrarily close to $a$ .
Effect of an open circle	Undefined there, unless a detached dot is plotted elsewhere.	Still exists, and equals the $y$ -coordinate of the open circle.
Role in the continuity test	Condition 1: the value must be defined.	Condition 2: the two-sided limit must exist.

Common Misconception Check

Myth 1: "If a function is undefined at $x = a$ , it has no predictable behavior near $x = a$ ."
- Truth: A limit never samples the point. Take $f(x) = \frac{x^2 - 4}{x - 2}$ . The function has no value at $x = 2$ , yet for inputs such as $1.999$ and $2.001$ the outputs sit at $3.999$ and $4.001$ . The behavior near the point is well defined and equals $4$ .
Myth 2: "The solid dot on a graph decides what the limit must equal."
- Truth: A solid dot reports where the function lands; a limit reports where the curve was heading. If both branches of a piecewise function meet at height $3$ and a solid dot sits at $(a, 7)$ , the limit is $3$ and the function value is $7$ .
Myth 3: "Behavior near a point is the average of the two sides."
- Truth: A limit is a trend, not an arithmetic mean. If the left side approaches $5$ and the right side approaches $1$ , the two-sided limit does not exist. The number $3$ is the average of the two sides, and the function never approaches it.

Topic-Specific Study Method

The Punctured-Neighborhood Protocol

Borrowed idea: The printed value at $a$ is placed where the eye lands first. The limit is scored from the two approaches, not from that dot. Remove $x = a$ from the workspace before reading anything. Cross out a table row at $a$ , cover a plotted dot, ignore a branch written for $x = a$ . What remains is a punctured neighborhood. The limit is the height both sides share there. Students lose the point by reading the dot first and then fitting the arms to it.

Representation on the Exam	How the Trap Is Set	Removal Step	What to Write or Select
Piecewise graph	An open circle at $(2, 4)$ with a solid dot at $(2, -1)$ .	Lay the pencil vertically along $x = 2$ and trace both branches into it.	The limit is $4$ , the height of the open circle. The dot is $f(2)$ .
Data table	Rows for $x = 2.999$ and $x = 3.001$ , plus a row at $x = 3$ reading $100$ .	Draw a vertical line through the $x = 3$ column and read only the innermost pair.	$\lim_{x \to 3} f(x) = 6$ , and $f(3) = 100$ is reported separately.
Piecewise formula	$f(x) = x^2 + 3$ for $x \neq 1$ , with $f(1) = 0$ .	Set aside the branch written for $x = 1$ before substituting.	$\lim_{x \to 1} f(x) = 1 + 3 = 4$ , while $f(1) = 0$ .
Absolute-value quotient	$f(x) = \dfrac{\lvert x - 5 \rvert}{x - 5}$ , undefined at $x = 5$ .	Test one input on each side: $x = 4.9$ gives $-1$ , $x = 5.1$ gives $1$ .	The sides disagree, so the limit does not exist.

Executable Action Steps: ① Run the pencil-shield habit on graphs. On every graph item, lay the pencil along $x = a$ before reading the choices. Trace each branch into the pencil and name the height they meet at. Then uncover the dot and write $f(a)$ on a separate line. Do this on twenty graphs. ② Slash the center column on tables. Whenever a table lists $x = a$ , draw an X through that column at once, then circle the two entries closest to $a$ on either side. Base the estimate on those two circles only. ③ Write the two-side proof on free-response items. Draft $\lim_{x \to a^-} f(x) = L_1$ and $\lim_{x \to a^+} f(x) = L_2$ on separate lines, then state whether $L_1 = L_2$ before writing anything about continuity.

Training & Error Diagnosis

Core Archetype (Simplest Form): Let $f$ be the function defined by $f(x) = \begin{cases} 4 - x^2, & x \neq 1 \cr 5, & x = 1 \end{cases}$ Determine $\lim_{x \to 1} f(x)$ and $f(1)$ , then state whether the behavior near $x = 1$ matches the behavior at $x = 1$ .

Standard Execution:

Near the point. Limits inspect inputs with $x \neq 1$ , so use the rule $f(x) = 4 - x^2$ : $\lim_{x \to 1} f(x) = \lim_{x \to 1} (4 - x^2) = 4 - 1 = 3$

At the point. The definition gives $f(1) = 5$ directly.

Compare. The limit is $3$ and the value is $5$ . Since $3 \neq 5$ , the two facts disagree, and the disagreement is a removable discontinuity at $x = 1$ .

Exam Variation (Context / Parameter Shift): A function $g$ is continuous for every real number except $x = -2$ . The table records values of $g$ in a punctured neighborhood of $-2$ .

$x$	$-2.01$	$-2.001$	$-2.0001$	$-2$	$-1.9999$	$-1.999$	$-1.99$
$g(x)$	$4.0401$	$4.0040$	$4.0004$	$\text{undefined}$	$3.9996$	$3.9960$	$3.9601$

Find the best estimate for $\lim_{x \to -2} g(x)$ . If $g(-2)$ were defined as $4$ , how would the continuity of $g$ at $x = -2$ change?

Solution: Reading inward on the left, the values run $4.0401, 4.0040, 4.0004$ , so $g(x) \to 4$ as $x \to -2^-$ . Reading inward on the right, they run $3.9996, 3.9960, 3.9601$ in reverse, so $g(x) \to 4$ as $x \to -2^+$ . Both sides crowd $4$ , so $\lim_{x \to -2} g(x) = 4$ . Defining $g(-2) = 4$ would make the value match the limit, which removes the discontinuity and makes $g$ continuous at $x = -2$ .

Common Traps & Failure Signals:

The plotted-dot answer (Signal: a graph with both an open circle and a detached solid dot at $x = a$ ): The dot supplies $f(a)$ and the open circle supplies the limit. Any choice equal to the dot has answered the wrong question.
The center-column answer (Signal: a table entry at $x = a$ that breaks the numerical pattern): The out-of-pattern number is placed there to be chosen. Slash the column and estimate from the flanking rows.
The one-side assumption (Signal: a table or graph that supplies data on one side of $a$ only): One side is not a verdict. A table with inputs below $a$ only supports a left-hand limit, so the two-sided limit stays undetermined.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Point-versus-Neighborhood Confusion $\rightarrow$ Draw a vertical line at $x = a$ , inspect only the approaching branches, and record $f(a)$ on a separate line.
One-Side Reading Neglect: You checked only one side of the punctured neighborhood.
Justification Collapse: Because you cited the point value $f(a)$ in place of evaluating the two-sided approach across the punctured neighborhood, you forfeited the justification point even though your final number happened to match the key.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No scored FRQ part in any released exam; the reading step behind every graph-based item
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Graph readings that compare the trend near a point with the printed value
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Graphical Decoupling of Limit and Point Value

The graph of $f$ shows a curve whose two branches close in on the open circle at $(3, 2)$ . A solid dot is plotted at $(3, -4)$ .

What is the value of $\lim_{x \to 3} f(x)$ ?

MCQ 2

Piecewise Rule With a Printed Value

Let $h$ be the function defined by

$h(x) = \begin{cases} \dfrac{x^2 - 9}{x - 3}, & x \neq 3 \cr 1, & x = 3 \end{cases}$

Which statement correctly describes the behavior of $h$ at and near $x = 3$ ?

MCQ 3

Table With an Out-of-Pattern Center Entry

The table gives values of a function $f$ for inputs near $0$ .

$x$	$-0.1$	$-0.01$	$-0.001$	$0$	$0.001$	$0.01$	$0.1$
$f(x)$	$0.99$	$0.9999$	$0.999999$	$5$	$0.999999$	$0.9999$	$0.99$

Which is the best estimate of $\lim_{x \to 0} f(x)$ ?

FRQ 4

Multi-Part Analysis of Behavior Near a Point

Consider the function $f$ defined by

$f(x) = \begin{cases} 2x + k, & x < 2 \cr 8, & x = 2 \cr x^2 - 2x + 4, & x > 2 \end{cases}$

where $k$ is a real constant.

(a) Evaluate $\lim_{x \to 2^+} f(x)$ and state the value of $f(2)$ .

(b) Find the value of $k$ for which the left and right behaviors of $f$ near $x = 2$ agree. Show the setup.

(c) Using that value of $k$ , determine whether $f$ is continuous at $x = 2$ . Justify your answer.

Cheatsheet