Unit 1Three Common Types of Non-Existent Limits (DNE)

Concept 4.120Built for May 2027 AP Exam

Three Common Types of Non-Existent Limits (DNE)

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

The three common DNE cases are the ones where a function fails to settle on a single finite value as $x$ nears a specific input. Unbounded behaviour near a vertical asymptote, constant oscillation that never converges, and mismatched one-sided limits at a jump cover almost every non-existent limit on the exam. The trap is that a limit written as $\lim_{x \to a} f(x) = \infty$ looks like an answer: infinity is not a finite value, so that limit still does not exist. You will be asked to categorise why a limit fails, from a graph, an equation, or a table of values.

On the AP Exam

This concept lands on both sections. On multiple choice you identify whether a limit exists from a graph, a table, or a function equation. On free response the requirement is stronger: you must name the DNE type in the justification, because the justification point is tied to the cause, not the conclusion.

What you’ll see: Limit expressions, piecewise function graphs, and discontinuous curve plots.
What you’ll do: Identify DNE limits, justify the claim, and compute the one-sided values that back it up.
Memory hook: JIO — Jump (unequal sides), Infinite (unbounded), Oscillating (no convergence). And the positive half of the rule: $L = R$ and both finite means the limit exists — even at a hole.
Use it on the exam: For MCQs: Match the graph, function, or table to one of the three types, or rule out DNE entirely for a removable discontinuity to kill the distractor set. For FRQs: Name the cause explicitly and in the rubric's own words — a bare “the limit does not exist” earns nothing when a justification is required.

Concept Review

Core Rule for Limit Existence

A limit exists only if both one-sided limits are finite and equal. Anything else is DNE.

Limit exists: Both one-sided limits are finite and equal to the same value.
Limit DNE: Fails that requirement — one side infinite, sides unequal, or no settling at all.

The Three Types

Each has a distinct identifying feature. Naming the wrong one costs the justification point even when the conclusion is right.

J — Jump. Left and right one-sided limits are finite but different.
I — Infinite. The function runs to $+\infty$ or $-\infty$ from at least one side of the point.
O — Oscillating. The function oscillates without converging to a single finite value near the point.

Figure 1 — The three failure shapes, side by side.

Caption. Jump: both sides reach a finite height, but not the same one. Infinite: the curve leaves the frame, so there is no height to compare. Oscillating: the swing keeps its full amplitude no matter how close you get, so no single height is ever settled on.

Three panels: a jump between two finite heights, an unbounded curve at a vertical asymptote, and a constant-amplitude oscillation

Distinguishing the Three

Look at	Jump	Infinite	Oscillating
Failure trigger	Unequal finite one-sided limits.	Approaches $\pm\infty$ from at least one side.	Never converges to one finite value.
Graphical cue	A vertical step between finite $y$ -values at $x=a$ .	The curve shoots toward $\pm\infty$ and forms a vertical asymptote.	Tight, constant-amplitude up-and-down motion near $x=a$ .
Standard example	A piecewise function with mismatched left and right values.	$f(x) = \dfrac{1}{x}$ at $x = 0$ .	$\sin\left(\dfrac{1}{x}\right)$ at $x = 0$ .

Two Corrections Worth Memorising

A limit at a removable discontinuity
- Correct: it exists, even when the function is undefined at the point.
- Common error: claiming DNE because there is a hole.
An infinite limit, $\lim_{x \to a} f(x) = \infty$
- Correct: classified as DNE.
- Common error: treating infinity as a value the limit attains.

Topic-Specific Study Method

The JIO Sort

Borrowed idea: A jump hands you two finite numbers to compare side by side. An infinite case gives you two arrows where the numbers should sit. An oscillation gives you no settled height at all, however far you zoom in. Ask those three questions in that order before your pen reaches the answer line. Students lose the point by writing the verdict first and then hunting for evidence that fits it, which is backwards, because the rubric scores the cause rather than the conclusion. Cause first, verdict second.

Read both sides as numbers. If you can write a finite number for each side, you are in J or in "exists". Compare them.
If a side has no number, ask which kind of no. Arrows to $\pm\infty$ means I. No settled height at all means O.
Check for damping before you commit to O. If the swing amplitude shrinks toward zero, the limit exists. $x\sin(1/x)$ is not $\sin(1/x)$ .
Write the phrase, not the label. "The one-sided limits are unequal" scores; "J" does not.

Representation on the Exam	How the Trap Is Set	The Sort Step	What to Write or Select
Graph with a step	Two arms end at different heights, both filled or both open.	Both sides are finite, so compare them directly.	DNE — the finite one-sided limits are unequal.
Graph rising off the top	$\infty$ is printed as a choice, inviting you to pick it.	No finite height on at least one side — this is I, not an answer.	DNE — unbounded near the point.
Formula with $\sin(1/x)$	The graph is unreadable if you try to eyeball it.	Check the amplitude of the swing, not the swing itself.	DNE if the amplitude holds; a finite value if it damps out.
Table with a jump row	Neighbouring rows sit far apart and the centre row is refused.	Compare the inner pair on the left with the inner pair on the right.	DNE — the two sides settle on different values.
FRQ justification	"Does not exist" alone is worth nothing.	Pick the cause first, then attach the matching sentence.	See the justification phrases in the cheatsheet below.

Training & Error Diagnosis

Core Archetype (Simplest Form): Let $g$ be the function defined by $g(x) = \begin{cases} 2x + 1, & x < 3 \cr 7, & x = 3 \cr x^{2} + 1, & x > 3 \end{cases}$ Determine whether $\lim_{x \to 3} g(x)$ exists.

Standard Execution:

Left side. For $x < 3$ , $g(x) = 2x + 1$ , so $\lim_{x \to 3^-} g(x) = 7$ .

Right side. For $x > 3$ , $g(x) = x^{2} + 1$ , so $\lim_{x \to 3^+} g(x) = 10$ .

Compare. Both are finite but $7 \neq 10$ , so the two-sided limit does not exist. The cause is a jump. Note that $g(3) = 7$ is irrelevant; quoting it earns nothing.

Exam Variation (Context / Parameter Shift): Let $h(x) = \dfrac{1}{(x-3)^{2}}$ and $k(x) = (x-3)\sin\left(\dfrac{1}{x-3}\right)$ for $x \neq 3$ . Classify each limit at $x = 3$ .

Solution: As $x \to 3$ from either side, $(x-3)^{2} \to 0^{+}$ , so $h(x) \to +\infty$ . Unbounded means the limit does not exist, and the cause is I. For $k$ , since $-1 \le \sin\!\left(\frac{1}{x-3}\right) \le 1$ , we get $-|x-3| \le k(x) \le |x-3|$ . Both bounds go to $0$ , so by the squeeze theorem $\lim_{x \to 3} k(x) = 0$ . The oscillation is damped: the amplitude $|x-3|$ shrinks to zero, so this is not the oscillating DNE case.

Common Traps & Failure Signals:

The unjustified generic DNE claim (Signal: the answer is a bare "does not exist"): The justification point is tied to naming the cause. A correct verdict with no cause earns nothing, and on multiple choice the wrong cause points at the wrong distractor.
Treating an unbounded limit as existent (Signal: $\infty$ is offered as a choice next to DNE): Infinity notation describes direction, not a value. When the question asks whether a limit exists, the answer is DNE.
Assuming every discontinuity kills the limit (Signal: a graph with a hole): Compute the two one-sided limits first. Equal and finite means the limit exists, hole or not.
Calling a damped oscillation DNE (Signal: a function that wiggles but whose amplitude shrinks): Test the amplitude. Shrinking to zero means the limit exists and equals the central value.

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

Infinity-as-Answer Reflex: The arrow notation looks like a result, so the eye stops there and the item gets banked, even though neither side ever produced a finite number.
Discontinuity Equals DNE: You matched on the break.
Oscillation Overreach: You matched on the wiggle rather than on the trend of the amplitude, and that trend is the only thing separating a damped case from a real failure.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; No standalone item: the cause of failure gets named inside a limit item
10-Year Total Count: 1 of 9 Exams · 1 FRQ Part; Jump, unbounded run, and oscillation told apart in one line
FRQ Years Appeared: 2021 (Q4c)

Practice Examples

MCQ 1

Jump: The Midpoint Distractor

The graph of $f$ has two horizontal arms. The left arm ends at an open circle of height $1$ ; the right arm starts at an open circle of height $3$ . A solid dot is plotted at $(2, 2)$ .

What is $\lim_{x \to 2} f(x)$ ?

MCQ 2

Infinity Is Not an Answer

For $f(x) = \dfrac{1}{x^{2}}$ , determine whether $\lim_{x \to 0} f(x)$ exists.

MCQ 3

Damped Oscillation Still Has a Limit

Evaluate $\lim_{x \to 0} x\sin\left(\dfrac{1}{x}\right)$ .

FRQ 4

Classify Three Failures and One Non-Failure

Let $g(x) = \begin{cases} 2x + 1, & x < 3 \cr 7, & x = 3 \cr x^{2} + 1, & x > 3 \end{cases}$

Also let $h(x) = \dfrac{1}{(x-3)^{2}}$ and $k(x) = (x-3)\sin\left(\dfrac{1}{x-3}\right)$ for $x \neq 3$ .

(a) Find $\lim_{x \to 3^{-}} g(x)$ and $\lim_{x \to 3^{+}} g(x)$ .

(b) Does $\lim_{x \to 3} g(x)$ exist? Justify your answer.

(c) Does $\lim_{x \to 3} h(x)$ exist? Justify your answer by naming the type.

(d) Does $\lim_{x \to 3} k(x)$ exist? Justify your answer by naming the type.

Cheatsheet

Naming the DNE Type

	Jump	Infinite	Oscillating
Trigger	Finite one-sided limits that disagree.	$\pm\infty$ from at least one side.	No convergence to one finite value.
Cue	Vertical step between two heights.	Curve leaves the frame; vertical asymptote.	Tight constant-amplitude wiggle.
Example	Mismatched piecewise rules.	$1/x$ at $0$ .	$\sin(1/x)$ at $0$ .
FRQ phrase	The limit does not exist because the finite one-sided limits $\lim_{x \to a^{-}} f(x)$ and $\lim_{x \to a^{+}} f(x)$ are unequal.	The limit does not exist because the function is unbounded near $x = a$ .	The limit does not exist because the function oscillates indefinitely and has no single finite limit near $x = a$ .
Correction	—	$\lim_{x \to a} f(x) = \infty$ describes behaviour; it is not a finite limit, so the limit does not exist.	A damped oscillation such as $x\sin(1/x)$ at $0$ does have a limit, because the amplitude shrinks to zero.

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Unit 1Three Common Types of Non-Existent Limits (DNE)

Concept 4.120Built for May 2027 AP Exam

Three Common Types of Non-Existent Limits (DNE)

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: Limit expressions, piecewise function graphs, and discontinuous curve plots.
What you’ll do: Identify DNE limits, justify the claim, and compute the one-sided values that back it up.
Memory hook: JIO — Jump (unequal sides), Infinite (unbounded), Oscillating (no convergence). And the positive half of the rule: $L = R$ and both finite means the limit exists — even at a hole.
Use it on the exam: For MCQs: Match the graph, function, or table to one of the three types, or rule out DNE entirely for a removable discontinuity to kill the distractor set. For FRQs: Name the cause explicitly and in the rubric's own words — a bare “the limit does not exist” earns nothing when a justification is required.

Concept Review

Core Rule for Limit Existence

A limit exists only if both one-sided limits are finite and equal. Anything else is DNE.

Limit exists: Both one-sided limits are finite and equal to the same value.
Limit DNE: Fails that requirement — one side infinite, sides unequal, or no settling at all.

The Three Types

Each has a distinct identifying feature. Naming the wrong one costs the justification point even when the conclusion is right.

J — Jump. Left and right one-sided limits are finite but different.
I — Infinite. The function runs to $+\infty$ or $-\infty$ from at least one side of the point.
O — Oscillating. The function oscillates without converging to a single finite value near the point.

Figure 1 — The three failure shapes, side by side.

Three panels: a jump between two finite heights, an unbounded curve at a vertical asymptote, and a constant-amplitude oscillation

Distinguishing the Three

Look at	Jump	Infinite	Oscillating
Failure trigger	Unequal finite one-sided limits.	Approaches $\pm\infty$ from at least one side.	Never converges to one finite value.
Graphical cue	A vertical step between finite $y$ -values at $x=a$ .	The curve shoots toward $\pm\infty$ and forms a vertical asymptote.	Tight, constant-amplitude up-and-down motion near $x=a$ .
Standard example	A piecewise function with mismatched left and right values.	$f(x) = \dfrac{1}{x}$ at $x = 0$ .	$\sin\left(\dfrac{1}{x}\right)$ at $x = 0$ .

Two Corrections Worth Memorising

A limit at a removable discontinuity
- Correct: it exists, even when the function is undefined at the point.
- Common error: claiming DNE because there is a hole.
An infinite limit, $\lim_{x \to a} f(x) = \infty$
- Correct: classified as DNE.
- Common error: treating infinity as a value the limit attains.

Topic-Specific Study Method

The JIO Sort

Read both sides as numbers. If you can write a finite number for each side, you are in J or in "exists". Compare them.
If a side has no number, ask which kind of no. Arrows to $\pm\infty$ means I. No settled height at all means O.
Check for damping before you commit to O. If the swing amplitude shrinks toward zero, the limit exists. $x\sin(1/x)$ is not $\sin(1/x)$ .
Write the phrase, not the label. "The one-sided limits are unequal" scores; "J" does not.

Representation on the Exam	How the Trap Is Set	The Sort Step	What to Write or Select
Graph with a step	Two arms end at different heights, both filled or both open.	Both sides are finite, so compare them directly.	DNE — the finite one-sided limits are unequal.
Graph rising off the top	$\infty$ is printed as a choice, inviting you to pick it.	No finite height on at least one side — this is I, not an answer.	DNE — unbounded near the point.
Formula with $\sin(1/x)$	The graph is unreadable if you try to eyeball it.	Check the amplitude of the swing, not the swing itself.	DNE if the amplitude holds; a finite value if it damps out.
Table with a jump row	Neighbouring rows sit far apart and the centre row is refused.	Compare the inner pair on the left with the inner pair on the right.	DNE — the two sides settle on different values.
FRQ justification	"Does not exist" alone is worth nothing.	Pick the cause first, then attach the matching sentence.	See the justification phrases in the cheatsheet below.

Training & Error Diagnosis

Standard Execution:

Left side. For $x < 3$ , $g(x) = 2x + 1$ , so $\lim_{x \to 3^-} g(x) = 7$ .

Right side. For $x > 3$ , $g(x) = x^{2} + 1$ , so $\lim_{x \to 3^+} g(x) = 10$ .

Compare. Both are finite but $7 \neq 10$ , so the two-sided limit does not exist. The cause is a jump. Note that $g(3) = 7$ is irrelevant; quoting it earns nothing.

Exam Variation (Context / Parameter Shift): Let $h(x) = \dfrac{1}{(x-3)^{2}}$ and $k(x) = (x-3)\sin\left(\dfrac{1}{x-3}\right)$ for $x \neq 3$ . Classify each limit at $x = 3$ .

Solution: As $x \to 3$ from either side, $(x-3)^{2} \to 0^{+}$ , so $h(x) \to +\infty$ . Unbounded means the limit does not exist, and the cause is I. For $k$ , since $-1 \le \sin\!\left(\frac{1}{x-3}\right) \le 1$ , we get $-|x-3| \le k(x) \le |x-3|$ . Both bounds go to $0$ , so by the squeeze theorem $\lim_{x \to 3} k(x) = 0$ . The oscillation is damped: the amplitude $|x-3|$ shrinks to zero, so this is not the oscillating DNE case.

Common Traps & Failure Signals:

The unjustified generic DNE claim (Signal: the answer is a bare "does not exist"): The justification point is tied to naming the cause. A correct verdict with no cause earns nothing, and on multiple choice the wrong cause points at the wrong distractor.
Treating an unbounded limit as existent (Signal: $\infty$ is offered as a choice next to DNE): Infinity notation describes direction, not a value. When the question asks whether a limit exists, the answer is DNE.
Assuming every discontinuity kills the limit (Signal: a graph with a hole): Compute the two one-sided limits first. Equal and finite means the limit exists, hole or not.
Calling a damped oscillation DNE (Signal: a function that wiggles but whose amplitude shrinks): Test the amplitude. Shrinking to zero means the limit exists and equals the central value.

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

Infinity-as-Answer Reflex: The arrow notation looks like a result, so the eye stops there and the item gets banked, even though neither side ever produced a finite number.
Discontinuity Equals DNE: You matched on the break.
Oscillation Overreach: You matched on the wiggle rather than on the trend of the amplitude, and that trend is the only thing separating a damped case from a real failure.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; No standalone item: the cause of failure gets named inside a limit item
10-Year Total Count: 1 of 9 Exams · 1 FRQ Part; Jump, unbounded run, and oscillation told apart in one line
FRQ Years Appeared: 2021 (Q4c)

Practice Examples

MCQ 1

Jump: The Midpoint Distractor

The graph of $f$ has two horizontal arms. The left arm ends at an open circle of height $1$ ; the right arm starts at an open circle of height $3$ . A solid dot is plotted at $(2, 2)$ .

What is $\lim_{x \to 2} f(x)$ ?

MCQ 2

Infinity Is Not an Answer

For $f(x) = \dfrac{1}{x^{2}}$ , determine whether $\lim_{x \to 0} f(x)$ exists.

MCQ 3

Damped Oscillation Still Has a Limit

Evaluate $\lim_{x \to 0} x\sin\left(\dfrac{1}{x}\right)$ .

FRQ 4

Classify Three Failures and One Non-Failure

Let $g(x) = \begin{cases} 2x + 1, & x < 3 \cr 7, & x = 3 \cr x^{2} + 1, & x > 3 \end{cases}$

Also let $h(x) = \dfrac{1}{(x-3)^{2}}$ and $k(x) = (x-3)\sin\left(\dfrac{1}{x-3}\right)$ for $x \neq 3$ .

(a) Find $\lim_{x \to 3^{-}} g(x)$ and $\lim_{x \to 3^{+}} g(x)$ .

(b) Does $\lim_{x \to 3} g(x)$ exist? Justify your answer.

(c) Does $\lim_{x \to 3} h(x)$ exist? Justify your answer by naming the type.

(d) Does $\lim_{x \to 3} k(x)$ exist? Justify your answer by naming the type.

Cheatsheet

Naming the DNE Type

	Jump	Infinite	Oscillating
Trigger	Finite one-sided limits that disagree.	$\pm\infty$ from at least one side.	No convergence to one finite value.
Cue	Vertical step between two heights.	Curve leaves the frame; vertical asymptote.	Tight constant-amplitude wiggle.
Example	Mismatched piecewise rules.	$1/x$ at $0$ .	$\sin(1/x)$ at $0$ .
FRQ phrase	The limit does not exist because the finite one-sided limits $\lim_{x \to a^{-}} f(x)$ and $\lim_{x \to a^{+}} f(x)$ are unequal.	The limit does not exist because the function is unbounded near $x = a$ .	The limit does not exist because the function oscillates indefinitely and has no single finite limit near $x = a$ .
Correction	—	$\lim_{x \to a} f(x) = \infty$ describes behaviour; it is not a finite limit, so the limit does not exist.	A damped oscillation such as $x\sin(1/x)$ at $0$ does have a limit, because the amplitude shrinks to zero.

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Start Topic Practice Next Topic: 1.9 Algebraic Properties of Limits

Unit 1Three Common Types of Non-Existent Limits (DNE)

Concept 4.120Built for May 2027 AP Exam

Three Common Types of Non-Existent Limits (DNE)

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: Limit expressions, piecewise function graphs, and discontinuous curve plots.
What you’ll do: Identify DNE limits, justify the claim, and compute the one-sided values that back it up.
Memory hook: JIO — Jump (unequal sides), Infinite (unbounded), Oscillating (no convergence). And the positive half of the rule: $L = R$ and both finite means the limit exists — even at a hole.
Use it on the exam: For MCQs: Match the graph, function, or table to one of the three types, or rule out DNE entirely for a removable discontinuity to kill the distractor set. For FRQs: Name the cause explicitly and in the rubric's own words — a bare “the limit does not exist” earns nothing when a justification is required.

Concept Review

Core Rule for Limit Existence

A limit exists only if both one-sided limits are finite and equal. Anything else is DNE.

Limit exists: Both one-sided limits are finite and equal to the same value.
Limit DNE: Fails that requirement — one side infinite, sides unequal, or no settling at all.

The Three Types

Each has a distinct identifying feature. Naming the wrong one costs the justification point even when the conclusion is right.

J — Jump. Left and right one-sided limits are finite but different.
I — Infinite. The function runs to $+\infty$ or $-\infty$ from at least one side of the point.
O — Oscillating. The function oscillates without converging to a single finite value near the point.

Figure 1 — The three failure shapes, side by side.

Three panels: a jump between two finite heights, an unbounded curve at a vertical asymptote, and a constant-amplitude oscillation

Distinguishing the Three

Look at	Jump	Infinite	Oscillating
Failure trigger	Unequal finite one-sided limits.	Approaches $\pm\infty$ from at least one side.	Never converges to one finite value.
Graphical cue	A vertical step between finite $y$ -values at $x=a$ .	The curve shoots toward $\pm\infty$ and forms a vertical asymptote.	Tight, constant-amplitude up-and-down motion near $x=a$ .
Standard example	A piecewise function with mismatched left and right values.	$f(x) = \dfrac{1}{x}$ at $x = 0$ .	$\sin\left(\dfrac{1}{x}\right)$ at $x = 0$ .

Two Corrections Worth Memorising

A limit at a removable discontinuity
- Correct: it exists, even when the function is undefined at the point.
- Common error: claiming DNE because there is a hole.
An infinite limit, $\lim_{x \to a} f(x) = \infty$
- Correct: classified as DNE.
- Common error: treating infinity as a value the limit attains.

Topic-Specific Study Method

The JIO Sort

Read both sides as numbers. If you can write a finite number for each side, you are in J or in "exists". Compare them.
If a side has no number, ask which kind of no. Arrows to $\pm\infty$ means I. No settled height at all means O.
Check for damping before you commit to O. If the swing amplitude shrinks toward zero, the limit exists. $x\sin(1/x)$ is not $\sin(1/x)$ .
Write the phrase, not the label. "The one-sided limits are unequal" scores; "J" does not.

Representation on the Exam	How the Trap Is Set	The Sort Step	What to Write or Select
Graph with a step	Two arms end at different heights, both filled or both open.	Both sides are finite, so compare them directly.	DNE — the finite one-sided limits are unequal.
Graph rising off the top	$\infty$ is printed as a choice, inviting you to pick it.	No finite height on at least one side — this is I, not an answer.	DNE — unbounded near the point.
Formula with $\sin(1/x)$	The graph is unreadable if you try to eyeball it.	Check the amplitude of the swing, not the swing itself.	DNE if the amplitude holds; a finite value if it damps out.
Table with a jump row	Neighbouring rows sit far apart and the centre row is refused.	Compare the inner pair on the left with the inner pair on the right.	DNE — the two sides settle on different values.
FRQ justification	"Does not exist" alone is worth nothing.	Pick the cause first, then attach the matching sentence.	See the justification phrases in the cheatsheet below.

Training & Error Diagnosis

Standard Execution:

Left side. For $x < 3$ , $g(x) = 2x + 1$ , so $\lim_{x \to 3^-} g(x) = 7$ .

Right side. For $x > 3$ , $g(x) = x^{2} + 1$ , so $\lim_{x \to 3^+} g(x) = 10$ .

Compare. Both are finite but $7 \neq 10$ , so the two-sided limit does not exist. The cause is a jump. Note that $g(3) = 7$ is irrelevant; quoting it earns nothing.

Exam Variation (Context / Parameter Shift): Let $h(x) = \dfrac{1}{(x-3)^{2}}$ and $k(x) = (x-3)\sin\left(\dfrac{1}{x-3}\right)$ for $x \neq 3$ . Classify each limit at $x = 3$ .

Solution: As $x \to 3$ from either side, $(x-3)^{2} \to 0^{+}$ , so $h(x) \to +\infty$ . Unbounded means the limit does not exist, and the cause is I. For $k$ , since $-1 \le \sin\!\left(\frac{1}{x-3}\right) \le 1$ , we get $-|x-3| \le k(x) \le |x-3|$ . Both bounds go to $0$ , so by the squeeze theorem $\lim_{x \to 3} k(x) = 0$ . The oscillation is damped: the amplitude $|x-3|$ shrinks to zero, so this is not the oscillating DNE case.

Common Traps & Failure Signals:

The unjustified generic DNE claim (Signal: the answer is a bare "does not exist"): The justification point is tied to naming the cause. A correct verdict with no cause earns nothing, and on multiple choice the wrong cause points at the wrong distractor.
Treating an unbounded limit as existent (Signal: $\infty$ is offered as a choice next to DNE): Infinity notation describes direction, not a value. When the question asks whether a limit exists, the answer is DNE.
Assuming every discontinuity kills the limit (Signal: a graph with a hole): Compute the two one-sided limits first. Equal and finite means the limit exists, hole or not.
Calling a damped oscillation DNE (Signal: a function that wiggles but whose amplitude shrinks): Test the amplitude. Shrinking to zero means the limit exists and equals the central value.

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

Infinity-as-Answer Reflex: The arrow notation looks like a result, so the eye stops there and the item gets banked, even though neither side ever produced a finite number.
Discontinuity Equals DNE: You matched on the break.
Oscillation Overreach: You matched on the wiggle rather than on the trend of the amplitude, and that trend is the only thing separating a damped case from a real failure.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; No standalone item: the cause of failure gets named inside a limit item
10-Year Total Count: 1 of 9 Exams · 1 FRQ Part; Jump, unbounded run, and oscillation told apart in one line
FRQ Years Appeared: 2021 (Q4c)

Practice Examples

MCQ 1

Jump: The Midpoint Distractor

The graph of $f$ has two horizontal arms. The left arm ends at an open circle of height $1$ ; the right arm starts at an open circle of height $3$ . A solid dot is plotted at $(2, 2)$ .

What is $\lim_{x \to 2} f(x)$ ?

MCQ 2

Infinity Is Not an Answer

For $f(x) = \dfrac{1}{x^{2}}$ , determine whether $\lim_{x \to 0} f(x)$ exists.

MCQ 3

Damped Oscillation Still Has a Limit

Evaluate $\lim_{x \to 0} x\sin\left(\dfrac{1}{x}\right)$ .

FRQ 4

Classify Three Failures and One Non-Failure

Let $g(x) = \begin{cases} 2x + 1, & x < 3 \cr 7, & x = 3 \cr x^{2} + 1, & x > 3 \end{cases}$

Also let $h(x) = \dfrac{1}{(x-3)^{2}}$ and $k(x) = (x-3)\sin\left(\dfrac{1}{x-3}\right)$ for $x \neq 3$ .

(a) Find $\lim_{x \to 3^{-}} g(x)$ and $\lim_{x \to 3^{+}} g(x)$ .

(b) Does $\lim_{x \to 3} g(x)$ exist? Justify your answer.

(c) Does $\lim_{x \to 3} h(x)$ exist? Justify your answer by naming the type.

(d) Does $\lim_{x \to 3} k(x)$ exist? Justify your answer by naming the type.

Cheatsheet

Naming the DNE Type

	Jump	Infinite	Oscillating
Trigger	Finite one-sided limits that disagree.	$\pm\infty$ from at least one side.	No convergence to one finite value.
Cue	Vertical step between two heights.	Curve leaves the frame; vertical asymptote.	Tight constant-amplitude wiggle.
Example	Mismatched piecewise rules.	$1/x$ at $0$ .	$\sin(1/x)$ at $0$ .
FRQ phrase	The limit does not exist because the finite one-sided limits $\lim_{x \to a^{-}} f(x)$ and $\lim_{x \to a^{+}} f(x)$ are unequal.	The limit does not exist because the function is unbounded near $x = a$ .	The limit does not exist because the function oscillates indefinitely and has no single finite limit near $x = a$ .
Correction	—	$\lim_{x \to a} f(x) = \infty$ describes behaviour; it is not a finite limit, so the limit does not exist.	A damped oscillation such as $x\sin(1/x)$ at $0$ does have a limit, because the amplitude shrinks to zero.

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