Unit 1Behavior Near a Point

Concept 4.130Built for May 2027 AP Exam

Behavior Near a Point

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Behavior near a point is a description, not a number. In a narrow window around $a$ , a function can settle on one height, run off toward $+\infty$ or $-\infty$ , or refuse to settle at all by oscillating. Each description carries its own vocabulary, and exam prompts reward the precise one. "Approaches $3$ from below" and "approaches $3$ from above" are different statements even though both name the same limit. "Increases without bound" is not the same as "the limit is infinity," because no limit exists in that case. Reading local behavior well means answering three questions in order: does the output settle, if so at what height and from which side, and if not, how does it fail.

On the AP Exam

AP Calculus BC tests this skill through the representations and reasoning described below.

What you’ll see: A graph zoomed to a small window, a formula with a denominator that vanishes, or a table whose entries grow, and a prompt asking you to describe rather than compute.
What you’ll do: Classify the local behavior first, then attach the precise phrase: settles from below, settles from above, increases without bound, decreases without bound, or oscillates.
Memory hook: Settle, run, or wobble.
Use it on the exam: For MCQs: Match the wording to the picture before doing any algebra. An option that says "the limit is $\infty$ " is wrong whenever the outputs grow without bound, because no limit exists in that case. For FRQs: Describe each side separately when the sides differ. A one-line description of the whole neighborhood is only correct when both sides behave the same way.

Concept Review

The Three Local Modes

Figure 1 — A tolerance band around $L$ and a window around $a$ : the limit holds only when the curve stays inside both.

Caption. The curve must remain inside the horizontal band for every input inside the shaded column. Shrinking the band forces the column to shrink, and that trade is the entire content of the definition.

Near a point, exactly one of three things happens.

The outputs settle. There is a single height that both arms approach, either from above, from below, or from opposite sides. The limit exists.
The outputs run. At least one arm grows without bound in one direction. No limit exists, though the divergence has a describable direction.
The outputs wobble. The values keep moving between heights no matter how narrow the window becomes. No limit exists, and no direction describes the failure.

A curve crossing a shaded horizontal tolerance band and a shaded vertical input window whose overlap contains the marked point

The Vocabulary That Earns Points

"From below": Every output in the window is smaller than $L$ while the limit is $L$ . The curve climbs into the height.
"From above": Every output is larger than $L$ while the limit is $L$ . The curve descends into the height.
"Increases without bound": Outputs eventually exceed any number you name. The limit does not exist; the phrase records how it fails.
"Oscillates": The outputs revisit a range of heights indefinitely. No limit exists and no infinity describes the failure.

Reading Local Behavior in Four Steps

Fix the window. Everything is judged inside a narrow band around $a$ , with $a$ itself excluded.
Ask whether a single height appears. If both arms converge, record the height and stop.
If not, ask what the outputs do instead. Growing without bound, falling without bound, or staying bounded while never settling.
Attach direction and side. Say which arm does what, and whether the approach is from above or below.

Behavior Catalog

Local Behavior	What the Outputs Do	Limit	Useful Description
Settle from below	Climb toward $L$ while staying under it	$L$	Approaches $L$ from below
Settle from above	Descend toward $L$ while staying over it	$L$	Approaches $L$ from above
Settle from both sides	One arm from each side	$L$	Approaches $L$
Run up	Exceed every bound	DNE	Increases without bound
Run down	Fall below every bound	DNE	Decreases without bound
Wobble	Keep revisiting heights	DNE	Oscillates; no limit

Common Misconception Check

Myth 1: "If the outputs grow without bound, the limit is infinity."
- Truth: Infinity is not a number, so no limit exists. Writing $\lim = \infty$ is shorthand for the divergence, and multiple-choice options that treat $\infty$ as a value are traps.
Myth 2: "Approaching from below and approaching from above are the same thing."
- Truth: They name the same height but opposite geometry. Many rubrics award a point for the side of approach, and a response that omits it is incomplete even when the height is right.
Myth 3: "If the window is small enough, every function settles."
- Truth: Some functions oscillate in every window, no matter how narrow, because the oscillation frequency grows as the input approaches the target. Narrowing the window does not help there.

Topic-Specific Study Method

The Classify-Then-Describe Drill

Borrowed idea: Students rush into algebra when the prompt only wanted a description. The algebra then produces a number for a question that had no number as its answer. Classify first. Ask whether a single height appears, then ask how the outputs fail if none does. Name the mode, then name the side. Only after both names are on the page should any computation begin.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Denominator vanishing	A quotient whose bottom goes to zero	Does the top also vanish, or does the quotient run?	Check the numerator's limit first, then classify
Graph near a vertical line	A curve hugging one side of a line	Which direction, and from which side?	Name the side and the direction separately
Oscillating formula	A sine or cosine of a reciprocal	Does narrowing the window help?	Say oscillates, and do not quote a height
Approach from one side	A curve entirely above or below the height	Which side is the approach on?	State the height and the side in one sentence

Executable Action Steps: ① Write the mode before the number. Put one of three words on the page first: settle, run, or wobble. Every later sentence depends on that classification. ② Test the numerator and denominator separately. A quotient runs only when the bottom vanishes while the top does not. If both vanish, the behavior is undecided and needs more work. ③ Add the side of approach to every settling answer. "From below" or "from above" costs three words and frequently carries a rubric point.

Training & Error Diagnosis

Core Archetype (Simplest Form): Describe the behavior of $f(x) = \frac{1}{(x - 2)^2}$ as $x \to 2$ .

Standard Execution:

As $x \to 2$ , the denominator $(x - 2)^2 \to 0$ through positive values while the numerator stays at $1$ .

The quotient therefore grows without bound on both sides.

Because the denominator is squared, both arms run toward $+\infty$ , not opposite directions.

The limit does not exist; the correct description is that $f$ increases without bound as $x \to 2$ from either side.

Exam Variation (Context / Parameter Shift): 1. Opposite Directions. Describe the behavior of $g(x) = \frac{1}{x - 2}$ as $x \to 2$ .

Solution: From the left, $x - 2 < 0$ and small, so $g(x) \to -\infty$ . From the right, $x - 2 > 0$ and small, so $g(x) \to +\infty$ . The two arms run in opposite directions, so the two-sided limit does not exist and the sides must be described separately.

2. Wobble. Describe the behavior of $h(x) = \sin\left(\frac{1}{x}\right)$ as $x \to 0$ .

Solution: As $x$ approaches $0$ , the argument grows without bound and the sine cycles through every value between $-1$ and $1$ repeatedly. No height is approached, so the limit does not exist; the description is that $h$ oscillates and stays bounded.

Common Traps & Failure Signals:

The infinity-as-value slip (Signal: your final answer is $\infty$ on a question asking whether the limit exists): Growth without bound is a failure of the limit, not a value it takes.
The one-sided description applied to both sides (Signal: your sentence describes one arm and claims it for the pair): When the arms run in opposite directions, one description cannot cover both.
The premature cancellation (Signal: you simplified a quotient before checking the numerator's limit): Cancellation can hide a genuine divergence, especially when the canceled factor carries the sign.

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

Mode Skipped: You computed before classifying.
- Side Dropped: Because you recorded only the height the outputs approach, you left out whether they came in from above or from below, and the rubric line that asked for the side went unearned even though the height was correct.
- Window Trust: You assumed a small enough window would always reveal a single height, when some functions oscillate at every scale.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No scored FRQ part in any released BC exam asks for a pure local-behavior description; divergences that appear on free response arrive inside rate and graph items
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Stems that ask how a function fails rather than what value it reaches
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Classifying a Divergence

Let $f(x) = \frac{3}{(x - 4)^2}$ . Which describes the behavior of $f$ as $x \to 4$ ?

MCQ 2

Opposite Directions

Let $g(x) = \frac{1}{x - 3}$ . Which is true about the behavior of $g$ as $x \to 3$ ?

MCQ 3

Recognizing an Oscillation

Let $h(x) = \cos\left(\frac{1}{x}\right)$ for $x \neq 0$ . What happens to $h(x)$ as $x \to 0$ ?

FRQ 4

Three Functions, Three Local Behaviors

Consider the three functions defined for $x \neq 1$ : $f(x) = \frac{(x - 1)^2 - 4}{(x - 1)^2}, \qquad g(x) = \frac{2}{x - 1}, \qquad h(x) = (x - 1)\sin\left(\frac{1}{x - 1}\right)$

(a) Describe the behavior of $f$ as $x \to 1$ . State whether $\lim_{x \to 1} f(x)$ exists.

(b) Describe the behavior of $g$ as $x \to 1$ , treating the left and right sides separately. State whether $\lim_{x \to 1} g(x)$ exists.

(c) Describe the behavior of $h$ as $x \to 1$ . State whether $\lim_{x \to 1} h(x)$ exists, and justify your conclusion.

Cheatsheet

Core Reference

	Settles	Runs	Wobbles
Limit exists?	Yes	No	No
What you report	The height, plus the side	The direction of divergence	That no height is approached
Typical source	Continuous curve, hole	Vanishing denominator, no matching zero upstairs	Sine or cosine of a reciprocal
Exam phrase	Approaches $L$ from below	Increases without bound	Oscillates
Common error	Dropping the side	Calling $\infty$ a value	Quoting a value from one subsequence
Position in this course	Rest of Unit 1	Topic 1.20	Stated here, resolved by the squeeze theorem

Sentences That Score

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Unit 1Behavior Near a Point

Concept 4.130Built for May 2027 AP Exam

Behavior Near a Point

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus BC tests this skill through the representations and reasoning described below.

What you’ll see: A graph zoomed to a small window, a formula with a denominator that vanishes, or a table whose entries grow, and a prompt asking you to describe rather than compute.
What you’ll do: Classify the local behavior first, then attach the precise phrase: settles from below, settles from above, increases without bound, decreases without bound, or oscillates.
Memory hook: Settle, run, or wobble.
Use it on the exam: For MCQs: Match the wording to the picture before doing any algebra. An option that says "the limit is $\infty$ " is wrong whenever the outputs grow without bound, because no limit exists in that case. For FRQs: Describe each side separately when the sides differ. A one-line description of the whole neighborhood is only correct when both sides behave the same way.

Concept Review

The Three Local Modes

Figure 1 — A tolerance band around $L$ and a window around $a$ : the limit holds only when the curve stays inside both.

Near a point, exactly one of three things happens.

The outputs settle. There is a single height that both arms approach, either from above, from below, or from opposite sides. The limit exists.
The outputs run. At least one arm grows without bound in one direction. No limit exists, though the divergence has a describable direction.
The outputs wobble. The values keep moving between heights no matter how narrow the window becomes. No limit exists, and no direction describes the failure.

A curve crossing a shaded horizontal tolerance band and a shaded vertical input window whose overlap contains the marked point

The Vocabulary That Earns Points

"From below": Every output in the window is smaller than $L$ while the limit is $L$ . The curve climbs into the height.
"From above": Every output is larger than $L$ while the limit is $L$ . The curve descends into the height.
"Increases without bound": Outputs eventually exceed any number you name. The limit does not exist; the phrase records how it fails.
"Oscillates": The outputs revisit a range of heights indefinitely. No limit exists and no infinity describes the failure.

Reading Local Behavior in Four Steps

Fix the window. Everything is judged inside a narrow band around $a$ , with $a$ itself excluded.
Ask whether a single height appears. If both arms converge, record the height and stop.
If not, ask what the outputs do instead. Growing without bound, falling without bound, or staying bounded while never settling.
Attach direction and side. Say which arm does what, and whether the approach is from above or below.

Behavior Catalog

Local Behavior	What the Outputs Do	Limit	Useful Description
Settle from below	Climb toward $L$ while staying under it	$L$	Approaches $L$ from below
Settle from above	Descend toward $L$ while staying over it	$L$	Approaches $L$ from above
Settle from both sides	One arm from each side	$L$	Approaches $L$
Run up	Exceed every bound	DNE	Increases without bound
Run down	Fall below every bound	DNE	Decreases without bound
Wobble	Keep revisiting heights	DNE	Oscillates; no limit

Common Misconception Check

Myth 1: "If the outputs grow without bound, the limit is infinity."
- Truth: Infinity is not a number, so no limit exists. Writing $\lim = \infty$ is shorthand for the divergence, and multiple-choice options that treat $\infty$ as a value are traps.
Myth 2: "Approaching from below and approaching from above are the same thing."
- Truth: They name the same height but opposite geometry. Many rubrics award a point for the side of approach, and a response that omits it is incomplete even when the height is right.
Myth 3: "If the window is small enough, every function settles."
- Truth: Some functions oscillate in every window, no matter how narrow, because the oscillation frequency grows as the input approaches the target. Narrowing the window does not help there.

Topic-Specific Study Method

The Classify-Then-Describe Drill

Borrowed idea: Students rush into algebra when the prompt only wanted a description. The algebra then produces a number for a question that had no number as its answer. Classify first. Ask whether a single height appears, then ask how the outputs fail if none does. Name the mode, then name the side. Only after both names are on the page should any computation begin.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Denominator vanishing	A quotient whose bottom goes to zero	Does the top also vanish, or does the quotient run?	Check the numerator's limit first, then classify
Graph near a vertical line	A curve hugging one side of a line	Which direction, and from which side?	Name the side and the direction separately
Oscillating formula	A sine or cosine of a reciprocal	Does narrowing the window help?	Say oscillates, and do not quote a height
Approach from one side	A curve entirely above or below the height	Which side is the approach on?	State the height and the side in one sentence

Executable Action Steps: ① Write the mode before the number. Put one of three words on the page first: settle, run, or wobble. Every later sentence depends on that classification. ② Test the numerator and denominator separately. A quotient runs only when the bottom vanishes while the top does not. If both vanish, the behavior is undecided and needs more work. ③ Add the side of approach to every settling answer. "From below" or "from above" costs three words and frequently carries a rubric point.

Training & Error Diagnosis

Core Archetype (Simplest Form): Describe the behavior of $f(x) = \frac{1}{(x - 2)^2}$ as $x \to 2$ .

Standard Execution:

As $x \to 2$ , the denominator $(x - 2)^2 \to 0$ through positive values while the numerator stays at $1$ .

The quotient therefore grows without bound on both sides.

Because the denominator is squared, both arms run toward $+\infty$ , not opposite directions.

The limit does not exist; the correct description is that $f$ increases without bound as $x \to 2$ from either side.

Exam Variation (Context / Parameter Shift): 1. Opposite Directions. Describe the behavior of $g(x) = \frac{1}{x - 2}$ as $x \to 2$ .

Solution: From the left, $x - 2 < 0$ and small, so $g(x) \to -\infty$ . From the right, $x - 2 > 0$ and small, so $g(x) \to +\infty$ . The two arms run in opposite directions, so the two-sided limit does not exist and the sides must be described separately.

2. Wobble. Describe the behavior of $h(x) = \sin\left(\frac{1}{x}\right)$ as $x \to 0$ .

Solution: As $x$ approaches $0$ , the argument grows without bound and the sine cycles through every value between $-1$ and $1$ repeatedly. No height is approached, so the limit does not exist; the description is that $h$ oscillates and stays bounded.

Common Traps & Failure Signals:

The infinity-as-value slip (Signal: your final answer is $\infty$ on a question asking whether the limit exists): Growth without bound is a failure of the limit, not a value it takes.
The one-sided description applied to both sides (Signal: your sentence describes one arm and claims it for the pair): When the arms run in opposite directions, one description cannot cover both.
The premature cancellation (Signal: you simplified a quotient before checking the numerator's limit): Cancellation can hide a genuine divergence, especially when the canceled factor carries the sign.

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

Mode Skipped: You computed before classifying.
- Side Dropped: Because you recorded only the height the outputs approach, you left out whether they came in from above or from below, and the rubric line that asked for the side went unearned even though the height was correct.
- Window Trust: You assumed a small enough window would always reveal a single height, when some functions oscillate at every scale.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No scored FRQ part in any released BC exam asks for a pure local-behavior description; divergences that appear on free response arrive inside rate and graph items
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Stems that ask how a function fails rather than what value it reaches
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Classifying a Divergence

Let $f(x) = \frac{3}{(x - 4)^2}$ . Which describes the behavior of $f$ as $x \to 4$ ?

MCQ 2

Opposite Directions

Let $g(x) = \frac{1}{x - 3}$ . Which is true about the behavior of $g$ as $x \to 3$ ?

MCQ 3

Recognizing an Oscillation

Let $h(x) = \cos\left(\frac{1}{x}\right)$ for $x \neq 0$ . What happens to $h(x)$ as $x \to 0$ ?

FRQ 4

Three Functions, Three Local Behaviors

Consider the three functions defined for $x \neq 1$ : $f(x) = \frac{(x - 1)^2 - 4}{(x - 1)^2}, \qquad g(x) = \frac{2}{x - 1}, \qquad h(x) = (x - 1)\sin\left(\frac{1}{x - 1}\right)$

(a) Describe the behavior of $f$ as $x \to 1$ . State whether $\lim_{x \to 1} f(x)$ exists.

(b) Describe the behavior of $g$ as $x \to 1$ , treating the left and right sides separately. State whether $\lim_{x \to 1} g(x)$ exists.

(c) Describe the behavior of $h$ as $x \to 1$ . State whether $\lim_{x \to 1} h(x)$ exists, and justify your conclusion.

Cheatsheet

Core Reference

	Settles	Runs	Wobbles
Limit exists?	Yes	No	No
What you report	The height, plus the side	The direction of divergence	That no height is approached
Typical source	Continuous curve, hole	Vanishing denominator, no matching zero upstairs	Sine or cosine of a reciprocal
Exam phrase	Approaches $L$ from below	Increases without bound	Oscillates
Common error	Dropping the side	Calling $\infty$ a value	Quoting a value from one subsequence
Position in this course	Rest of Unit 1	Topic 1.20	Stated here, resolved by the squeeze theorem

Sentences That Score

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Start Topic Practice Next Topic: 1.7 Numerical Approach

Unit 1Behavior Near a Point

Concept 4.130Built for May 2027 AP Exam

Behavior Near a Point

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus BC tests this skill through the representations and reasoning described below.

What you’ll see: A graph zoomed to a small window, a formula with a denominator that vanishes, or a table whose entries grow, and a prompt asking you to describe rather than compute.
What you’ll do: Classify the local behavior first, then attach the precise phrase: settles from below, settles from above, increases without bound, decreases without bound, or oscillates.
Memory hook: Settle, run, or wobble.
Use it on the exam: For MCQs: Match the wording to the picture before doing any algebra. An option that says "the limit is $\infty$ " is wrong whenever the outputs grow without bound, because no limit exists in that case. For FRQs: Describe each side separately when the sides differ. A one-line description of the whole neighborhood is only correct when both sides behave the same way.

Concept Review

The Three Local Modes

Figure 1 — A tolerance band around $L$ and a window around $a$ : the limit holds only when the curve stays inside both.

Near a point, exactly one of three things happens.

The outputs settle. There is a single height that both arms approach, either from above, from below, or from opposite sides. The limit exists.
The outputs run. At least one arm grows without bound in one direction. No limit exists, though the divergence has a describable direction.
The outputs wobble. The values keep moving between heights no matter how narrow the window becomes. No limit exists, and no direction describes the failure.

A curve crossing a shaded horizontal tolerance band and a shaded vertical input window whose overlap contains the marked point

The Vocabulary That Earns Points

"From below": Every output in the window is smaller than $L$ while the limit is $L$ . The curve climbs into the height.
"From above": Every output is larger than $L$ while the limit is $L$ . The curve descends into the height.
"Increases without bound": Outputs eventually exceed any number you name. The limit does not exist; the phrase records how it fails.
"Oscillates": The outputs revisit a range of heights indefinitely. No limit exists and no infinity describes the failure.

Reading Local Behavior in Four Steps

Fix the window. Everything is judged inside a narrow band around $a$ , with $a$ itself excluded.
Ask whether a single height appears. If both arms converge, record the height and stop.
If not, ask what the outputs do instead. Growing without bound, falling without bound, or staying bounded while never settling.
Attach direction and side. Say which arm does what, and whether the approach is from above or below.

Behavior Catalog

Local Behavior	What the Outputs Do	Limit	Useful Description
Settle from below	Climb toward $L$ while staying under it	$L$	Approaches $L$ from below
Settle from above	Descend toward $L$ while staying over it	$L$	Approaches $L$ from above
Settle from both sides	One arm from each side	$L$	Approaches $L$
Run up	Exceed every bound	DNE	Increases without bound
Run down	Fall below every bound	DNE	Decreases without bound
Wobble	Keep revisiting heights	DNE	Oscillates; no limit

Common Misconception Check

Myth 1: "If the outputs grow without bound, the limit is infinity."
- Truth: Infinity is not a number, so no limit exists. Writing $\lim = \infty$ is shorthand for the divergence, and multiple-choice options that treat $\infty$ as a value are traps.
Myth 2: "Approaching from below and approaching from above are the same thing."
- Truth: They name the same height but opposite geometry. Many rubrics award a point for the side of approach, and a response that omits it is incomplete even when the height is right.
Myth 3: "If the window is small enough, every function settles."
- Truth: Some functions oscillate in every window, no matter how narrow, because the oscillation frequency grows as the input approaches the target. Narrowing the window does not help there.

Topic-Specific Study Method

The Classify-Then-Describe Drill

Borrowed idea: Students rush into algebra when the prompt only wanted a description. The algebra then produces a number for a question that had no number as its answer. Classify first. Ask whether a single height appears, then ask how the outputs fail if none does. Name the mode, then name the side. Only after both names are on the page should any computation begin.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Denominator vanishing	A quotient whose bottom goes to zero	Does the top also vanish, or does the quotient run?	Check the numerator's limit first, then classify
Graph near a vertical line	A curve hugging one side of a line	Which direction, and from which side?	Name the side and the direction separately
Oscillating formula	A sine or cosine of a reciprocal	Does narrowing the window help?	Say oscillates, and do not quote a height
Approach from one side	A curve entirely above or below the height	Which side is the approach on?	State the height and the side in one sentence

Executable Action Steps: ① Write the mode before the number. Put one of three words on the page first: settle, run, or wobble. Every later sentence depends on that classification. ② Test the numerator and denominator separately. A quotient runs only when the bottom vanishes while the top does not. If both vanish, the behavior is undecided and needs more work. ③ Add the side of approach to every settling answer. "From below" or "from above" costs three words and frequently carries a rubric point.

Training & Error Diagnosis

Core Archetype (Simplest Form): Describe the behavior of $f(x) = \frac{1}{(x - 2)^2}$ as $x \to 2$ .

Standard Execution:

As $x \to 2$ , the denominator $(x - 2)^2 \to 0$ through positive values while the numerator stays at $1$ .

The quotient therefore grows without bound on both sides.

Because the denominator is squared, both arms run toward $+\infty$ , not opposite directions.

The limit does not exist; the correct description is that $f$ increases without bound as $x \to 2$ from either side.

Exam Variation (Context / Parameter Shift): 1. Opposite Directions. Describe the behavior of $g(x) = \frac{1}{x - 2}$ as $x \to 2$ .

Solution: From the left, $x - 2 < 0$ and small, so $g(x) \to -\infty$ . From the right, $x - 2 > 0$ and small, so $g(x) \to +\infty$ . The two arms run in opposite directions, so the two-sided limit does not exist and the sides must be described separately.

2. Wobble. Describe the behavior of $h(x) = \sin\left(\frac{1}{x}\right)$ as $x \to 0$ .

Solution: As $x$ approaches $0$ , the argument grows without bound and the sine cycles through every value between $-1$ and $1$ repeatedly. No height is approached, so the limit does not exist; the description is that $h$ oscillates and stays bounded.

Common Traps & Failure Signals:

The infinity-as-value slip (Signal: your final answer is $\infty$ on a question asking whether the limit exists): Growth without bound is a failure of the limit, not a value it takes.
The one-sided description applied to both sides (Signal: your sentence describes one arm and claims it for the pair): When the arms run in opposite directions, one description cannot cover both.
The premature cancellation (Signal: you simplified a quotient before checking the numerator's limit): Cancellation can hide a genuine divergence, especially when the canceled factor carries the sign.

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

Mode Skipped: You computed before classifying.
- Side Dropped: Because you recorded only the height the outputs approach, you left out whether they came in from above or from below, and the rubric line that asked for the side went unearned even though the height was correct.
- Window Trust: You assumed a small enough window would always reveal a single height, when some functions oscillate at every scale.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No scored FRQ part in any released BC exam asks for a pure local-behavior description; divergences that appear on free response arrive inside rate and graph items
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Stems that ask how a function fails rather than what value it reaches
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Classifying a Divergence

Let $f(x) = \frac{3}{(x - 4)^2}$ . Which describes the behavior of $f$ as $x \to 4$ ?

MCQ 2

Opposite Directions

Let $g(x) = \frac{1}{x - 3}$ . Which is true about the behavior of $g$ as $x \to 3$ ?

MCQ 3

Recognizing an Oscillation

Let $h(x) = \cos\left(\frac{1}{x}\right)$ for $x \neq 0$ . What happens to $h(x)$ as $x \to 0$ ?

FRQ 4

Three Functions, Three Local Behaviors

Consider the three functions defined for $x \neq 1$ : $f(x) = \frac{(x - 1)^2 - 4}{(x - 1)^2}, \qquad g(x) = \frac{2}{x - 1}, \qquad h(x) = (x - 1)\sin\left(\frac{1}{x - 1}\right)$

(a) Describe the behavior of $f$ as $x \to 1$ . State whether $\lim_{x \to 1} f(x)$ exists.

(b) Describe the behavior of $g$ as $x \to 1$ , treating the left and right sides separately. State whether $\lim_{x \to 1} g(x)$ exists.

(c) Describe the behavior of $h$ as $x \to 1$ . State whether $\lim_{x \to 1} h(x)$ exists, and justify your conclusion.

Cheatsheet

Core Reference

	Settles	Runs	Wobbles
Limit exists?	Yes	No	No
What you report	The height, plus the side	The direction of divergence	That no height is approached
Typical source	Continuous curve, hole	Vanishing denominator, no matching zero upstairs	Sine or cosine of a reciprocal
Exam phrase	Approaches $L$ from below	Increases without bound	Oscillates
Common error	Dropping the side	Calling $\infty$ a value	Quoting a value from one subsequence
Position in this course	Rest of Unit 1	Topic 1.20	Stated here, resolved by the squeeze theorem

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