Unit 1Limit vs. Function Value

Concept 4.173Built for May 2027 AP Exam

Limit vs. Function Value

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Two questions sound identical and are not. "What is the limit as $x \to a$ ?" asks where the curve is heading. "What is $f(a)$ ?" asks where the dot actually is. They agree whenever the function is continuous at $a$ , and that agreement is so common that most students stop distinguishing the questions altogether, which is exactly what the exam tests. The limit survives a hole; the value does not. The value can sit far from the curve; the limit cannot. Reading a graph, a table, or a formula correctly requires asking which of the two quantities is being requested before doing anything else.

On the AP Exam

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

What you’ll see: A graph with an open circle and a displaced dot, a table whose middle row is defined differently from its neighbours, or a two-part question that asks for both quantities at the same point.
What you’ll do: Answer the two questions separately and explicitly, then state whether they agree and what that agreement or disagreement implies about continuity.
Memory hook: The limit is a destination; the value is a location.
Use it on the exam: For MCQs: Read the notation first. $\lim_{x \to a}$ and $f(a)$ are different symbols, and the distractors are built by answering the other one. For FRQs: When a stem asks for both, write both with their notation intact. A response that writes " $f(2) = 4$ " when the limit is $4$ and the value is $1$ earns nothing even though the number is present. On released BC papers: The distinction itself has never been the subject of a scored free-response part in the 2017–2026 BC sets, and that is unsurprising: free-response questions tend to ask for a value with a justification rather than for a contrast. What the released exams do show is that the distinction is load-bearing inside parts that read a graph. When a stem presents the graph of $f'$ and asks for $f'(2)$ , the answer comes from the plotted point, and when the same stem asks for a limit of a difference quotient, the answer comes from the surrounding behavior. Confusing the two produces a wrong number with a confident justification, which is the most expensive kind of error available on these exams.

Concept Review

Four Combinations, One Definition

Figure 1 — Limit and value are independent facts; continuity is the case where they coincide.

Caption. Only the first panel is continuity. The other three are distinct failures, and each one is a different answer to "what is happening at $a$ ".

Limit at $a$	Value at $a$	Relation	Diagnosis
Exists, equals $L$	Exists, equals $L$	They agree	Continuous at $a$
Exists, equals $L$	Does not exist	Cannot be compared	Removable hole
Exists, equals $L$	Exists, equals $M \neq L$	They disagree	Displaced point
Does not exist	Exists	Not applicable	Jump, asymptote, or oscillation

Four small panels comparing a continuous point, a hole, a displaced point, and a jump, each labelled with which combination of limit and value it represents

Reading a Graph

The graph encodes both quantities in different marks:

The open circle marks the height the curve approaches. That is the limit.
The filled dot marks the height the function actually attains. That is the value.
No mark at all at the approach point means the value does not exist, while the open circle still reports the limit.

A graph with an open circle at height $4$ and a filled dot at height $1$ , both above the same input, is telling you two different numbers on purpose. Neither is an error in the drawing.

Reading a Table

A table can present the same split. The column for $f(x)$ may show values converging to $4$ as $x$ approaches $2$ , while a footnote, a separate piecewise branch, or a bolded entry reports $f(2) = 1$ .

The habit that prevents errors is to record the two answers in two separate places on your page before comparing them. Students who write a single number lose the information the table was built to convey.

Why the Confusion Is So Persistent

Direct substitution works so often that it becomes an assumption rather than a technique. When substitution returns a number, the limit and the value coincide — because that is exactly what continuity means. The failure mode is generalising from that experience: students conclude that the limit is the value, and then apply the conclusion at points where the substitution principle was never licensed.

The corrective is to remember what each quantity is defined by. A limit is defined by values of $f$ near $a$ , excluding $a$ . A value is defined by $f$ at $a$ alone. Since the definitions use disjoint information, neither can determine the other without a continuity hypothesis.

Common Misconception Check

Myth 1: "The limit is the value the function reaches."
- Truth: The function may never reach it. The limit describes approach, and the approach can be toward a height the function never occupies.
Myth 2: "If $f(a)$ is undefined, the limit cannot exist."
- Truth: The limit ignores $a$ entirely. A hole at $a$ leaves the limit intact and removes only the value.
Myth 3: "If the limit and the value differ, one of them is wrong."
- Truth: They routinely differ, and that difference is the definition of a removable discontinuity rather than an inconsistency.

Topic-Specific Study Method

Answer Two Questions, Then Compare

Borrowed idea: A limit is a prediction, and a value is a fact. They usually agree, which is why the two get confused. They agree exactly when the function is continuous, and continuity is a theorem to check rather than a mood to sense. So the exam question is never which number the graph is heading toward; it is which of the two quantities is being requested. Read the notation before you read the graph. A limit asks where the curve is going, and a value asks where the dot is. Keep those two sentences separate and half the errors in this topic disappear, including the expensive one where a student reports a limit as though it were a function value.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Open circle with a displaced dot	Two different heights above one input	Which mark answers which question?	Open circle gives the limit, filled dot gives the value
Open circle alone	No filled dot at the input	Does the value exist at all?	Limit exists, value does not
Table with an odd middle row	Neighbouring rows consistent, centre row not	Are the two columns reporting different things?	Read the limit from the neighbours, the value from the centre
"Find $f(2)$ " then "find $\lim_{x \to 2} f(x)$ "	Two requests about one point	Can you keep the notation straight?	Answer separately, then state whether they agree
Piecewise with a redefined point	A single branch changed at $a$	Does redefining the dot move the curve?	The limit is unchanged; only the value moves

Executable Action Steps: ① Copy the notation exactly as written. If the question says $\lim_{x \to 2}$ , do not answer with $f(2)$ ; the symbols are the instruction. ② Write the two answers on separate lines before comparing them. Two lines force you to compute two things, which is the entire point of the item. ③ State the comparison in words. "The limit is $4$ and the value is $1$ , so they disagree and $f$ is not continuous at $2$ " is a complete answer; a bare number is not.

Training & Error Diagnosis

Core Archetype (Simplest Form): The graph of $f$ has an open circle at $(2, 4)$ and a filled dot at $(2, 1)$ . Find $\lim_{x \to 2} f(x)$ and $f(2)$ , and state whether $f$ is continuous at $x = 2$ .

Standard Execution:

The open circle reports the height the curve approaches, so $\lim_{x \to 2} f(x) = 4$ .

The filled dot reports the attained height, so $f(2) = 1$ .

Both exist and they differ, so clause three of the continuity definition fails.

The function is not continuous at $2$ , and the break is removable.

Exam Variation (Context / Parameter Shift): 1. From a Table. The table shows $f(1.9) = 3.98$ , $f(1.99) = 3.998$ , $f(2.01) = 4.002$ , $f(2.1) = 4.02$ , and a note says $f(2) = 1$ . Find the limit and the value at $2$ .

Solution: The neighbouring entries converge to $4$ , so the limit is $4$ . The note gives $f(2) = 1$ . The two disagree, so $f$ is not continuous at $2$ .

2. With a Parameter. Suppose $f(x) = \frac{x^2 - 9}{x - 3}$ for $x \neq 3$ and $f(3) = c$ . For which $c$ is $f$ continuous at $3$ ?

Solution: The limit is $6$ by cancellation, so continuity requires $c = 6$ . Any other value of $c$ leaves a displaced point.

Common Traps & Failure Signals:

The substituted answer (Signal: your response to a limit question cites $f(a)$ ): A limit at a point is computed from nearby values, never from the point itself.
The single number (Signal: one answer where the question asked for two things): Two quantities were requested; one line cannot contain both.
The unstated comparison (Signal: you found both numbers and stopped): The question usually also asks whether they agree, and that verdict is a separate point.

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

Notation Unread: You evaluated the function where the limit was asked.
- Assumption Made: Because you assumed a limit and a value must agree, you never tested continuity, and the substitution that had always worked silently supplied the wrong answer for a question about approach rather than attainment.
- Comparison Omitted: You produced both numbers correctly and never said whether they matched, leaving the actual question unanswered.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; Never the subject of a part; the distinction is load-bearing inside graph-reading items
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Zero parts, because free-response asks for a value with a justification, not for a contrast
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Reading the Two Marks

The graph of $f$ has a hole at $(3, 5)$ and a filled point at $(3, 2)$ . What are $\lim_{x \to 3} f(x)$ and $f(3)$ ?

MCQ 2

From Notation Alone

Suppose $\lim_{x \to 4} f(x) = 7$ and $f(4) = -1$ . Which statement is true?

MCQ 3

Which Question Was Asked?

A table shows values of $f$ approaching $12$ as $x$ approaches $5$ from both sides, and the table's center row reads "undefined". What is $\lim_{x \to 5} f(x)$ ?

FRQ 4

Two Quantities at One Point

Let $f$ be defined by $f(x) = \begin{cases} \dfrac{x^2 - 1}{x - 1} & x \neq 1 \cr 4 & x = 1 \end{cases}$

(a) Find $\lim_{x \to 1} f(x)$ , showing the algebraic step.

(b) State $f(1)$ and determine whether $f$ is continuous at $x = 1$ , justifying your answer with the three clauses.

(c) A student concludes from part (a) that $f(1) = 2$ . Identify the error in that inference and explain what information a limit does and does not supply.

Cheatsheet

Core Reference

	What It Reads	What It Ignores	When It Fails to Exist
$\lim_{x \to a} f(x)$	Values at inputs near $a$	The value at $a$ itself	One-sided limits differ, or a side is unbounded
$f(a)$	The definition at the single input $a$	Every other input	$a$ is outside the domain
Continuity at $a$	Both of the above	Nothing	Either fails, or they disagree

Sentences That Score

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Unit 1Limit vs. Function Value

Concept 4.173Built for May 2027 AP Exam

Limit vs. Function Value

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 with an open circle and a displaced dot, a table whose middle row is defined differently from its neighbours, or a two-part question that asks for both quantities at the same point.
What you’ll do: Answer the two questions separately and explicitly, then state whether they agree and what that agreement or disagreement implies about continuity.
Memory hook: The limit is a destination; the value is a location.
Use it on the exam: For MCQs: Read the notation first. $\lim_{x \to a}$ and $f(a)$ are different symbols, and the distractors are built by answering the other one. For FRQs: When a stem asks for both, write both with their notation intact. A response that writes " $f(2) = 4$ " when the limit is $4$ and the value is $1$ earns nothing even though the number is present. On released BC papers: The distinction itself has never been the subject of a scored free-response part in the 2017–2026 BC sets, and that is unsurprising: free-response questions tend to ask for a value with a justification rather than for a contrast. What the released exams do show is that the distinction is load-bearing inside parts that read a graph. When a stem presents the graph of $f'$ and asks for $f'(2)$ , the answer comes from the plotted point, and when the same stem asks for a limit of a difference quotient, the answer comes from the surrounding behavior. Confusing the two produces a wrong number with a confident justification, which is the most expensive kind of error available on these exams.

Concept Review

Four Combinations, One Definition

Figure 1 — Limit and value are independent facts; continuity is the case where they coincide.

Caption. Only the first panel is continuity. The other three are distinct failures, and each one is a different answer to "what is happening at $a$ ".

Limit at $a$	Value at $a$	Relation	Diagnosis
Exists, equals $L$	Exists, equals $L$	They agree	Continuous at $a$
Exists, equals $L$	Does not exist	Cannot be compared	Removable hole
Exists, equals $L$	Exists, equals $M \neq L$	They disagree	Displaced point
Does not exist	Exists	Not applicable	Jump, asymptote, or oscillation

Four small panels comparing a continuous point, a hole, a displaced point, and a jump, each labelled with which combination of limit and value it represents

Reading a Graph

The graph encodes both quantities in different marks:

The open circle marks the height the curve approaches. That is the limit.
The filled dot marks the height the function actually attains. That is the value.
No mark at all at the approach point means the value does not exist, while the open circle still reports the limit.

A graph with an open circle at height $4$ and a filled dot at height $1$ , both above the same input, is telling you two different numbers on purpose. Neither is an error in the drawing.

Reading a Table

Why the Confusion Is So Persistent

Common Misconception Check

Myth 1: "The limit is the value the function reaches."
- Truth: The function may never reach it. The limit describes approach, and the approach can be toward a height the function never occupies.
Myth 2: "If $f(a)$ is undefined, the limit cannot exist."
- Truth: The limit ignores $a$ entirely. A hole at $a$ leaves the limit intact and removes only the value.
Myth 3: "If the limit and the value differ, one of them is wrong."
- Truth: They routinely differ, and that difference is the definition of a removable discontinuity rather than an inconsistency.

Topic-Specific Study Method

Answer Two Questions, Then Compare

Borrowed idea: A limit is a prediction, and a value is a fact. They usually agree, which is why the two get confused. They agree exactly when the function is continuous, and continuity is a theorem to check rather than a mood to sense. So the exam question is never which number the graph is heading toward; it is which of the two quantities is being requested. Read the notation before you read the graph. A limit asks where the curve is going, and a value asks where the dot is. Keep those two sentences separate and half the errors in this topic disappear, including the expensive one where a student reports a limit as though it were a function value.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Open circle with a displaced dot	Two different heights above one input	Which mark answers which question?	Open circle gives the limit, filled dot gives the value
Open circle alone	No filled dot at the input	Does the value exist at all?	Limit exists, value does not
Table with an odd middle row	Neighbouring rows consistent, centre row not	Are the two columns reporting different things?	Read the limit from the neighbours, the value from the centre
"Find $f(2)$ " then "find $\lim_{x \to 2} f(x)$ "	Two requests about one point	Can you keep the notation straight?	Answer separately, then state whether they agree
Piecewise with a redefined point	A single branch changed at $a$	Does redefining the dot move the curve?	The limit is unchanged; only the value moves

Executable Action Steps: ① Copy the notation exactly as written. If the question says $\lim_{x \to 2}$ , do not answer with $f(2)$ ; the symbols are the instruction. ② Write the two answers on separate lines before comparing them. Two lines force you to compute two things, which is the entire point of the item. ③ State the comparison in words. "The limit is $4$ and the value is $1$ , so they disagree and $f$ is not continuous at $2$ " is a complete answer; a bare number is not.

Training & Error Diagnosis

Standard Execution:

The open circle reports the height the curve approaches, so $\lim_{x \to 2} f(x) = 4$ .

The filled dot reports the attained height, so $f(2) = 1$ .

Both exist and they differ, so clause three of the continuity definition fails.

The function is not continuous at $2$ , and the break is removable.

Solution: The neighbouring entries converge to $4$ , so the limit is $4$ . The note gives $f(2) = 1$ . The two disagree, so $f$ is not continuous at $2$ .

2. With a Parameter. Suppose $f(x) = \frac{x^2 - 9}{x - 3}$ for $x \neq 3$ and $f(3) = c$ . For which $c$ is $f$ continuous at $3$ ?

Solution: The limit is $6$ by cancellation, so continuity requires $c = 6$ . Any other value of $c$ leaves a displaced point.

Common Traps & Failure Signals:

The substituted answer (Signal: your response to a limit question cites $f(a)$ ): A limit at a point is computed from nearby values, never from the point itself.
The single number (Signal: one answer where the question asked for two things): Two quantities were requested; one line cannot contain both.
The unstated comparison (Signal: you found both numbers and stopped): The question usually also asks whether they agree, and that verdict is a separate point.

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

Notation Unread: You evaluated the function where the limit was asked.
- Assumption Made: Because you assumed a limit and a value must agree, you never tested continuity, and the substitution that had always worked silently supplied the wrong answer for a question about approach rather than attainment.
- Comparison Omitted: You produced both numbers correctly and never said whether they matched, leaving the actual question unanswered.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; Never the subject of a part; the distinction is load-bearing inside graph-reading items
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Zero parts, because free-response asks for a value with a justification, not for a contrast
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Reading the Two Marks

The graph of $f$ has a hole at $(3, 5)$ and a filled point at $(3, 2)$ . What are $\lim_{x \to 3} f(x)$ and $f(3)$ ?

MCQ 2

From Notation Alone

Suppose $\lim_{x \to 4} f(x) = 7$ and $f(4) = -1$ . Which statement is true?

MCQ 3

Which Question Was Asked?

A table shows values of $f$ approaching $12$ as $x$ approaches $5$ from both sides, and the table's center row reads "undefined". What is $\lim_{x \to 5} f(x)$ ?

FRQ 4

Two Quantities at One Point

Let $f$ be defined by $f(x) = \begin{cases} \dfrac{x^2 - 1}{x - 1} & x \neq 1 \cr 4 & x = 1 \end{cases}$

(a) Find $\lim_{x \to 1} f(x)$ , showing the algebraic step.

(b) State $f(1)$ and determine whether $f$ is continuous at $x = 1$ , justifying your answer with the three clauses.

(c) A student concludes from part (a) that $f(1) = 2$ . Identify the error in that inference and explain what information a limit does and does not supply.

Cheatsheet

Core Reference

	What It Reads	What It Ignores	When It Fails to Exist
$\lim_{x \to a} f(x)$	Values at inputs near $a$	The value at $a$ itself	One-sided limits differ, or a side is unbounded
$f(a)$	The definition at the single input $a$	Every other input	$a$ is outside the domain
Continuity at $a$	Both of the above	Nothing	Either fails, or they disagree

Sentences That Score

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Unit 1Limit vs. Function Value

Concept 4.173Built for May 2027 AP Exam

Limit vs. Function Value

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 with an open circle and a displaced dot, a table whose middle row is defined differently from its neighbours, or a two-part question that asks for both quantities at the same point.
What you’ll do: Answer the two questions separately and explicitly, then state whether they agree and what that agreement or disagreement implies about continuity.
Memory hook: The limit is a destination; the value is a location.
Use it on the exam: For MCQs: Read the notation first. $\lim_{x \to a}$ and $f(a)$ are different symbols, and the distractors are built by answering the other one. For FRQs: When a stem asks for both, write both with their notation intact. A response that writes " $f(2) = 4$ " when the limit is $4$ and the value is $1$ earns nothing even though the number is present. On released BC papers: The distinction itself has never been the subject of a scored free-response part in the 2017–2026 BC sets, and that is unsurprising: free-response questions tend to ask for a value with a justification rather than for a contrast. What the released exams do show is that the distinction is load-bearing inside parts that read a graph. When a stem presents the graph of $f'$ and asks for $f'(2)$ , the answer comes from the plotted point, and when the same stem asks for a limit of a difference quotient, the answer comes from the surrounding behavior. Confusing the two produces a wrong number with a confident justification, which is the most expensive kind of error available on these exams.

Concept Review

Four Combinations, One Definition

Figure 1 — Limit and value are independent facts; continuity is the case where they coincide.

Caption. Only the first panel is continuity. The other three are distinct failures, and each one is a different answer to "what is happening at $a$ ".

Limit at $a$	Value at $a$	Relation	Diagnosis
Exists, equals $L$	Exists, equals $L$	They agree	Continuous at $a$
Exists, equals $L$	Does not exist	Cannot be compared	Removable hole
Exists, equals $L$	Exists, equals $M \neq L$	They disagree	Displaced point
Does not exist	Exists	Not applicable	Jump, asymptote, or oscillation

Four small panels comparing a continuous point, a hole, a displaced point, and a jump, each labelled with which combination of limit and value it represents

Reading a Graph

The graph encodes both quantities in different marks:

The open circle marks the height the curve approaches. That is the limit.
The filled dot marks the height the function actually attains. That is the value.
No mark at all at the approach point means the value does not exist, while the open circle still reports the limit.

A graph with an open circle at height $4$ and a filled dot at height $1$ , both above the same input, is telling you two different numbers on purpose. Neither is an error in the drawing.

Reading a Table

Why the Confusion Is So Persistent

Common Misconception Check

Myth 1: "The limit is the value the function reaches."
- Truth: The function may never reach it. The limit describes approach, and the approach can be toward a height the function never occupies.
Myth 2: "If $f(a)$ is undefined, the limit cannot exist."
- Truth: The limit ignores $a$ entirely. A hole at $a$ leaves the limit intact and removes only the value.
Myth 3: "If the limit and the value differ, one of them is wrong."
- Truth: They routinely differ, and that difference is the definition of a removable discontinuity rather than an inconsistency.

Topic-Specific Study Method

Answer Two Questions, Then Compare

Borrowed idea: A limit is a prediction, and a value is a fact. They usually agree, which is why the two get confused. They agree exactly when the function is continuous, and continuity is a theorem to check rather than a mood to sense. So the exam question is never which number the graph is heading toward; it is which of the two quantities is being requested. Read the notation before you read the graph. A limit asks where the curve is going, and a value asks where the dot is. Keep those two sentences separate and half the errors in this topic disappear, including the expensive one where a student reports a limit as though it were a function value.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Open circle with a displaced dot	Two different heights above one input	Which mark answers which question?	Open circle gives the limit, filled dot gives the value
Open circle alone	No filled dot at the input	Does the value exist at all?	Limit exists, value does not
Table with an odd middle row	Neighbouring rows consistent, centre row not	Are the two columns reporting different things?	Read the limit from the neighbours, the value from the centre
"Find $f(2)$ " then "find $\lim_{x \to 2} f(x)$ "	Two requests about one point	Can you keep the notation straight?	Answer separately, then state whether they agree
Piecewise with a redefined point	A single branch changed at $a$	Does redefining the dot move the curve?	The limit is unchanged; only the value moves

Executable Action Steps: ① Copy the notation exactly as written. If the question says $\lim_{x \to 2}$ , do not answer with $f(2)$ ; the symbols are the instruction. ② Write the two answers on separate lines before comparing them. Two lines force you to compute two things, which is the entire point of the item. ③ State the comparison in words. "The limit is $4$ and the value is $1$ , so they disagree and $f$ is not continuous at $2$ " is a complete answer; a bare number is not.

Training & Error Diagnosis

Standard Execution:

The open circle reports the height the curve approaches, so $\lim_{x \to 2} f(x) = 4$ .

The filled dot reports the attained height, so $f(2) = 1$ .

Both exist and they differ, so clause three of the continuity definition fails.

The function is not continuous at $2$ , and the break is removable.

Solution: The neighbouring entries converge to $4$ , so the limit is $4$ . The note gives $f(2) = 1$ . The two disagree, so $f$ is not continuous at $2$ .

2. With a Parameter. Suppose $f(x) = \frac{x^2 - 9}{x - 3}$ for $x \neq 3$ and $f(3) = c$ . For which $c$ is $f$ continuous at $3$ ?

Solution: The limit is $6$ by cancellation, so continuity requires $c = 6$ . Any other value of $c$ leaves a displaced point.

Common Traps & Failure Signals:

The substituted answer (Signal: your response to a limit question cites $f(a)$ ): A limit at a point is computed from nearby values, never from the point itself.
The single number (Signal: one answer where the question asked for two things): Two quantities were requested; one line cannot contain both.
The unstated comparison (Signal: you found both numbers and stopped): The question usually also asks whether they agree, and that verdict is a separate point.

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

Notation Unread: You evaluated the function where the limit was asked.
- Assumption Made: Because you assumed a limit and a value must agree, you never tested continuity, and the substitution that had always worked silently supplied the wrong answer for a question about approach rather than attainment.
- Comparison Omitted: You produced both numbers correctly and never said whether they matched, leaving the actual question unanswered.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; Never the subject of a part; the distinction is load-bearing inside graph-reading items
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Zero parts, because free-response asks for a value with a justification, not for a contrast
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Reading the Two Marks

The graph of $f$ has a hole at $(3, 5)$ and a filled point at $(3, 2)$ . What are $\lim_{x \to 3} f(x)$ and $f(3)$ ?

MCQ 2

From Notation Alone

Suppose $\lim_{x \to 4} f(x) = 7$ and $f(4) = -1$ . Which statement is true?

MCQ 3

Which Question Was Asked?

A table shows values of $f$ approaching $12$ as $x$ approaches $5$ from both sides, and the table's center row reads "undefined". What is $\lim_{x \to 5} f(x)$ ?

FRQ 4

Two Quantities at One Point

Let $f$ be defined by $f(x) = \begin{cases} \dfrac{x^2 - 1}{x - 1} & x \neq 1 \cr 4 & x = 1 \end{cases}$

(a) Find $\lim_{x \to 1} f(x)$ , showing the algebraic step.

(b) State $f(1)$ and determine whether $f$ is continuous at $x = 1$ , justifying your answer with the three clauses.

(c) A student concludes from part (a) that $f(1) = 2$ . Identify the error in that inference and explain what information a limit does and does not supply.

Cheatsheet

Core Reference

	What It Reads	What It Ignores	When It Fails to Exist
$\lim_{x \to a} f(x)$	Values at inputs near $a$	The value at $a$ itself	One-sided limits differ, or a side is unbounded
$f(a)$	The definition at the single input $a$	Every other input	$a$ is outside the domain
Continuity at $a$	Both of the above	Nothing	Either fails, or they disagree

Sentences That Score

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