Unit 1Graphical Analysis

Concept 4.267Built for May 2027 AP Exam

Graphical Analysis

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

A graph is a table of limits drawn in ink. At each marked input you read the two arms, compare their heights, and report one of three verdicts: the arms meet at a finite height, they meet at different heights, or at least one arm runs off the page. The function value shown as a solid dot is separate evidence and never overrides the arms. Read one target input at a time, because a single picture usually contains several features and each one asks its own question. Where a curve passes through cleanly, the limit and the value agree. Where a hole sits at one height and a dot sits at another, the limit follows the hole. Where the curve hugs a vertical line, the limit fails and you say so with the direction of the blow-up.

On the AP Exam

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

What you’ll see: A single figure with two or three marked inputs, each carrying a different feature, followed by several one-line questions or a matching set of statements.
What you’ll do: Take each marked input in turn, read both arms without looking at the dot, and write the verdict in symbols.
Memory hook: Arms decide. Dots only report.
Use it on the exam: For MCQs: Answer the specific input asked, not the most visually dramatic feature. A graph with an asymptote on the right often has a plain question about a hole on the left. For FRQs: Name the input, the two one-sided heights, and the verdict in that order. A justification that skips the one-sided values loses the reasoning point even when the verdict is right.

Concept Review

Three Features, Three Questions

Figure 1 — The same curve read at three different inputs.

Caption. At $a_1$ the arms pass through one height, so the limit equals the value. At $a_2$ the arms end at $2.98$ and $1.72$ , so the limit fails. At $a_3$ the arms meet at the hole while the dot sits higher, so the limit exists and differs from the value.

A curve that crosses smoothly at a sub one, jumps down at a sub two, and has a hollow circle with a displaced filled dot at a sub three

What Each Feature Tells You

Clean crossing: Both arms arrive at the same height and the curve is drawn through. The limit exists and equals the function value there.
Jump: Both arms arrive at finite but different heights. Each one-sided limit exists; the two-sided limit does not.
Hole with a displaced dot: Both arms arrive at the same height, but the point is drawn elsewhere. The limit exists and equals the hole's height, not the dot's.
Vertical asymptote: At least one arm runs off the page. The limit fails, and the direction of the divergence is worth recording.

The Five-Step Graph Read

Find every marked input on the horizontal axis. Those are the only inputs any question will ask about.
Read the left arm at that input. Follow the drawn curve in from smaller inputs and note the height it approaches, whether or not a dot is present.
Read the right arm the same way. Do not let the left answer influence the right reading.
Compare the two heights. Equal finite heights give the limit; otherwise write DNE and say why.
Check the dot only if the question mentions $f(a)$ . Otherwise it is decoration.

Verdict Table

Feature on the Graph	Left Arm	Right Arm	Limit	Value $f(a)$
Clean crossing	Height $L$	Height $L$	$L$	$L$
Jump	Height $L_1$	Height $L_2 \neq L_1$	DNE	Whichever dot is filled
Hole, dot elsewhere	Height $L$	Height $L$	$L$	The dot's height
Vertical asymptote	$\pm\infty$ or finite	Finite or $\pm\infty$	DNE	Usually undefined

Common Misconception Check

Myth 1: "The limit at a hole does not exist because the curve is broken."
- Truth: A removable hole leaves both arms converging on the same height. The limit exists at every hole of that kind, which is why the topic is called removable.
Myth 2: "A dot above the hole means the limit is the dot's height."
- Truth: The dot reports the function value. The limit is whatever the arms converge on, and the exam draws these apart precisely to test the distinction.
Myth 3: "If the graph is not defined to the left of a point, the limit does not exist."
- Truth: At an endpoint of the domain only one arm exists, and the one-sided limit is the whole story. Existence then rests on that single side.

Topic-Specific Study Method

The One-Input Sweep

Borrowed idea: Why does the same student who reads a hole correctly then misread a jump two inches to its right? Because they read the whole picture at once and let the most dramatic feature set the tone. Read the graph once per input instead, circling each marked input and answering only the feature inside the circle. One circle, one verdict.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Two marks, two features	A hole on the left, a jump on the right	Which input does this particular question name?	Answer only the named input, then move to the next question
Statement matching	Four sentences, one graph	Which sentence survives every feature?	Test each sentence at one specific input, not in general
Endpoint item	A curve starting at the left edge	What can be said when only one arm exists?	Read the single arm and call it a one-sided limit
Dot-versus-hole item	Both a hollow circle and a filled dot on one vertical line	Which height is the limit and which is the value?	Arms for the limit, dot for the value, both reported

Executable Action Steps: ① Number the marked inputs before reading anything. Write $a_1, a_2, a_3$ along the axis and put a small tick beside each. Numbering forces you to treat them as separate questions. ② Say the two arm heights out loud, in order. "Left arm reaches three, right arm starts at two." The spoken pair is harder to fudge than a glance. ③ Write DNE with a reason attached. A bare DNE is worth partial credit at best; "DNE because the one-sided limits are three and two" is worth full credit.

Training & Error Diagnosis

Core Archetype (Simplest Form): The graph of $f$ consists of the line $y = x + 1$ for $x < 2$ and the line $y = 2x - 1$ for $x > 2$ , with $f(2) = 3$ marked by a filled dot. Find $\lim_{x \to 2^-} f(x)$ , $\lim_{x \to 2^+} f(x)$ , and $\lim_{x \to 2} f(x)$ .

Standard Execution:

Left arm: as $x \to 2^-$ , $y = x + 1 \to 3$ .

Right arm: as $x \to 2^+$ , $y = 2x - 1 \to 3$ .

The two arms agree at $3$ , so $\lim_{x \to 2} f(x) = 3$ .

The dot also sits at $3$ , so this graph happens to be continuous at $x = 2$ , even though the rule changes there.

Exam Variation (Context / Parameter Shift): 1. Hole With a Displaced Dot. A curve approaches height $4$ from both sides at $x = 5$ , but a filled dot sits at $(5, 1)$ . Find $\lim_{x \to 5} f(x)$ and $f(5)$ .

Solution: The two arms converge on $4$ , so the limit is $4$ . The dot reports $f(5) = 1$ . Both answers are correct and they disagree, which is allowed.

2. Endpoint Item. A function is defined only for $x \ge 0$ , and its graph rises to height $2$ as $x \to 0^+$ . What is $\lim_{x \to 0} f(x)$ ?

Solution: Only the right arm exists, and it reaches $2$ . The limit is $2$ , read as the one-sided limit from within the domain. There is nothing to compare on the left.

Common Traps & Failure Signals:

The drama effect (Signal: your answer describes the asymptote when the question names a hole): Visual weight is not the question. Read the input the stem names.
The dot override (Signal: your limit equals the filled dot's height while the arms disagree): The dot is a value, not an approach. If the arms say $4$ and the dot says $1$ , the limit is $4$ .
The averaged jump (Signal: your limit sits between the two arm heights): A two-sided limit requires equality. Halfway is not a height the arms reach.

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

Feature Blending: You merged three questions into one impression.
- Dot Deference: Because the filled dot was drawn darker and larger than the hollow circle right beside it, you treated it as the authoritative answer and reported a function value where the question had asked for an approach height.
- Unstated Verdict: You wrote DNE without naming the two one-sided heights that produced it.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No scored FRQ part in any released BC exam is a pure graph-reading limit item; graphical stems that do appear are read for derivatives or accumulations
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Multi-feature figures where several limits are requested from one picture
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Reading a Removable Hole

The graph of $f$ has a hollow circle at $(3, 6)$ approached by both arms, and a filled dot at $(3, 2)$ . What is $\lim_{x \to 3} f(x)$ ?

MCQ 2

Reading a Jump

The graph of $g$ approaches height $5$ from the left at $x = 1$ and height $-2$ from the right at $x = 1$ . What is $\lim_{x \to 1} g(x)$ ?

MCQ 3

Endpoint Reading

The graph of $h$ is drawn only for $x \ge 2$ , and as $x \to 2^+$ the curve rises to height $7$ . Which statement is true?

FRQ 4

A Piecewise Graph Read at Three Inputs

Let $f$ be defined by $f(x) = \begin{cases} 2 - x, & x < 1 \cr 4, & x = 1 \cr x + 3, & 1 < x < 4 \cr 10 - x, & x \ge 4 \end{cases}$

(a) Find $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$ . Does $\lim_{x \to 1} f(x)$ exist? Justify your answer.

(b) Find $\lim_{x \to 4^-} f(x)$ and $\lim_{x \to 4^+} f(x)$ . Does $\lim_{x \to 4} f(x)$ exist? Justify your answer.

(c) At which of $x = 1$ and $x = 4$ does the limit fail, and at which does the function value differ from the limit? State each answer separately.

Cheatsheet

Core Reference

	What to Look At	What Decides the Answer
Target input	The number named in the question	Not the most eye-catching feature
Left-hand limit	The curve arriving from smaller inputs	Height reached by that arm
Right-hand limit	The curve arriving from larger inputs	Height reached by that arm
Two-sided limit	Both arms together	Equality of two finite heights
Function value	The filled dot on the vertical line	Its height, and nothing else
Hole	A hollow circle where the arms meet	The hole's height is the limit
Jump	Two arms ending at different heights	DNE, with both heights quoted
Asymptote	A curve hugging a vertical line	DNE, with the direction of divergence

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Unit 1Graphical Analysis

Concept 4.267Built for May 2027 AP Exam

Graphical Analysis

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 single figure with two or three marked inputs, each carrying a different feature, followed by several one-line questions or a matching set of statements.
What you’ll do: Take each marked input in turn, read both arms without looking at the dot, and write the verdict in symbols.
Memory hook: Arms decide. Dots only report.
Use it on the exam: For MCQs: Answer the specific input asked, not the most visually dramatic feature. A graph with an asymptote on the right often has a plain question about a hole on the left. For FRQs: Name the input, the two one-sided heights, and the verdict in that order. A justification that skips the one-sided values loses the reasoning point even when the verdict is right.

Concept Review

Three Features, Three Questions

Figure 1 — The same curve read at three different inputs.

A curve that crosses smoothly at a sub one, jumps down at a sub two, and has a hollow circle with a displaced filled dot at a sub three

What Each Feature Tells You

Clean crossing: Both arms arrive at the same height and the curve is drawn through. The limit exists and equals the function value there.
Jump: Both arms arrive at finite but different heights. Each one-sided limit exists; the two-sided limit does not.
Hole with a displaced dot: Both arms arrive at the same height, but the point is drawn elsewhere. The limit exists and equals the hole's height, not the dot's.
Vertical asymptote: At least one arm runs off the page. The limit fails, and the direction of the divergence is worth recording.

The Five-Step Graph Read

Find every marked input on the horizontal axis. Those are the only inputs any question will ask about.
Read the left arm at that input. Follow the drawn curve in from smaller inputs and note the height it approaches, whether or not a dot is present.
Read the right arm the same way. Do not let the left answer influence the right reading.
Compare the two heights. Equal finite heights give the limit; otherwise write DNE and say why.
Check the dot only if the question mentions $f(a)$ . Otherwise it is decoration.

Verdict Table

Feature on the Graph	Left Arm	Right Arm	Limit	Value $f(a)$
Clean crossing	Height $L$	Height $L$	$L$	$L$
Jump	Height $L_1$	Height $L_2 \neq L_1$	DNE	Whichever dot is filled
Hole, dot elsewhere	Height $L$	Height $L$	$L$	The dot's height
Vertical asymptote	$\pm\infty$ or finite	Finite or $\pm\infty$	DNE	Usually undefined

Common Misconception Check

Myth 1: "The limit at a hole does not exist because the curve is broken."
- Truth: A removable hole leaves both arms converging on the same height. The limit exists at every hole of that kind, which is why the topic is called removable.
Myth 2: "A dot above the hole means the limit is the dot's height."
- Truth: The dot reports the function value. The limit is whatever the arms converge on, and the exam draws these apart precisely to test the distinction.
Myth 3: "If the graph is not defined to the left of a point, the limit does not exist."
- Truth: At an endpoint of the domain only one arm exists, and the one-sided limit is the whole story. Existence then rests on that single side.

Topic-Specific Study Method

The One-Input Sweep

Borrowed idea: Why does the same student who reads a hole correctly then misread a jump two inches to its right? Because they read the whole picture at once and let the most dramatic feature set the tone. Read the graph once per input instead, circling each marked input and answering only the feature inside the circle. One circle, one verdict.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Two marks, two features	A hole on the left, a jump on the right	Which input does this particular question name?	Answer only the named input, then move to the next question
Statement matching	Four sentences, one graph	Which sentence survives every feature?	Test each sentence at one specific input, not in general
Endpoint item	A curve starting at the left edge	What can be said when only one arm exists?	Read the single arm and call it a one-sided limit
Dot-versus-hole item	Both a hollow circle and a filled dot on one vertical line	Which height is the limit and which is the value?	Arms for the limit, dot for the value, both reported

Executable Action Steps: ① Number the marked inputs before reading anything. Write $a_1, a_2, a_3$ along the axis and put a small tick beside each. Numbering forces you to treat them as separate questions. ② Say the two arm heights out loud, in order. "Left arm reaches three, right arm starts at two." The spoken pair is harder to fudge than a glance. ③ Write DNE with a reason attached. A bare DNE is worth partial credit at best; "DNE because the one-sided limits are three and two" is worth full credit.

Training & Error Diagnosis

Standard Execution:

Left arm: as $x \to 2^-$ , $y = x + 1 \to 3$ .

Right arm: as $x \to 2^+$ , $y = 2x - 1 \to 3$ .

The two arms agree at $3$ , so $\lim_{x \to 2} f(x) = 3$ .

The dot also sits at $3$ , so this graph happens to be continuous at $x = 2$ , even though the rule changes there.

Solution: The two arms converge on $4$ , so the limit is $4$ . The dot reports $f(5) = 1$ . Both answers are correct and they disagree, which is allowed.

2. Endpoint Item. A function is defined only for $x \ge 0$ , and its graph rises to height $2$ as $x \to 0^+$ . What is $\lim_{x \to 0} f(x)$ ?

Solution: Only the right arm exists, and it reaches $2$ . The limit is $2$ , read as the one-sided limit from within the domain. There is nothing to compare on the left.

Common Traps & Failure Signals:

The drama effect (Signal: your answer describes the asymptote when the question names a hole): Visual weight is not the question. Read the input the stem names.
The dot override (Signal: your limit equals the filled dot's height while the arms disagree): The dot is a value, not an approach. If the arms say $4$ and the dot says $1$ , the limit is $4$ .
The averaged jump (Signal: your limit sits between the two arm heights): A two-sided limit requires equality. Halfway is not a height the arms reach.

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

Feature Blending: You merged three questions into one impression.
- Dot Deference: Because the filled dot was drawn darker and larger than the hollow circle right beside it, you treated it as the authoritative answer and reported a function value where the question had asked for an approach height.
- Unstated Verdict: You wrote DNE without naming the two one-sided heights that produced it.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No scored FRQ part in any released BC exam is a pure graph-reading limit item; graphical stems that do appear are read for derivatives or accumulations
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Multi-feature figures where several limits are requested from one picture
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Reading a Removable Hole

The graph of $f$ has a hollow circle at $(3, 6)$ approached by both arms, and a filled dot at $(3, 2)$ . What is $\lim_{x \to 3} f(x)$ ?

MCQ 2

Reading a Jump

The graph of $g$ approaches height $5$ from the left at $x = 1$ and height $-2$ from the right at $x = 1$ . What is $\lim_{x \to 1} g(x)$ ?

MCQ 3

Endpoint Reading

The graph of $h$ is drawn only for $x \ge 2$ , and as $x \to 2^+$ the curve rises to height $7$ . Which statement is true?

FRQ 4

A Piecewise Graph Read at Three Inputs

Let $f$ be defined by $f(x) = \begin{cases} 2 - x, & x < 1 \cr 4, & x = 1 \cr x + 3, & 1 < x < 4 \cr 10 - x, & x \ge 4 \end{cases}$

(a) Find $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$ . Does $\lim_{x \to 1} f(x)$ exist? Justify your answer.

(b) Find $\lim_{x \to 4^-} f(x)$ and $\lim_{x \to 4^+} f(x)$ . Does $\lim_{x \to 4} f(x)$ exist? Justify your answer.

(c) At which of $x = 1$ and $x = 4$ does the limit fail, and at which does the function value differ from the limit? State each answer separately.

Cheatsheet

Core Reference

	What to Look At	What Decides the Answer
Target input	The number named in the question	Not the most eye-catching feature
Left-hand limit	The curve arriving from smaller inputs	Height reached by that arm
Right-hand limit	The curve arriving from larger inputs	Height reached by that arm
Two-sided limit	Both arms together	Equality of two finite heights
Function value	The filled dot on the vertical line	Its height, and nothing else
Hole	A hollow circle where the arms meet	The hole's height is the limit
Jump	Two arms ending at different heights	DNE, with both heights quoted
Asymptote	A curve hugging a vertical line	DNE, with the direction of divergence

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Unit 1Graphical Analysis

Concept 4.267Built for May 2027 AP Exam

Graphical Analysis

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 single figure with two or three marked inputs, each carrying a different feature, followed by several one-line questions or a matching set of statements.
What you’ll do: Take each marked input in turn, read both arms without looking at the dot, and write the verdict in symbols.
Memory hook: Arms decide. Dots only report.
Use it on the exam: For MCQs: Answer the specific input asked, not the most visually dramatic feature. A graph with an asymptote on the right often has a plain question about a hole on the left. For FRQs: Name the input, the two one-sided heights, and the verdict in that order. A justification that skips the one-sided values loses the reasoning point even when the verdict is right.

Concept Review

Three Features, Three Questions

Figure 1 — The same curve read at three different inputs.

A curve that crosses smoothly at a sub one, jumps down at a sub two, and has a hollow circle with a displaced filled dot at a sub three

What Each Feature Tells You

Clean crossing: Both arms arrive at the same height and the curve is drawn through. The limit exists and equals the function value there.
Jump: Both arms arrive at finite but different heights. Each one-sided limit exists; the two-sided limit does not.
Hole with a displaced dot: Both arms arrive at the same height, but the point is drawn elsewhere. The limit exists and equals the hole's height, not the dot's.
Vertical asymptote: At least one arm runs off the page. The limit fails, and the direction of the divergence is worth recording.

The Five-Step Graph Read

Find every marked input on the horizontal axis. Those are the only inputs any question will ask about.
Read the left arm at that input. Follow the drawn curve in from smaller inputs and note the height it approaches, whether or not a dot is present.
Read the right arm the same way. Do not let the left answer influence the right reading.
Compare the two heights. Equal finite heights give the limit; otherwise write DNE and say why.
Check the dot only if the question mentions $f(a)$ . Otherwise it is decoration.

Verdict Table

Feature on the Graph	Left Arm	Right Arm	Limit	Value $f(a)$
Clean crossing	Height $L$	Height $L$	$L$	$L$
Jump	Height $L_1$	Height $L_2 \neq L_1$	DNE	Whichever dot is filled
Hole, dot elsewhere	Height $L$	Height $L$	$L$	The dot's height
Vertical asymptote	$\pm\infty$ or finite	Finite or $\pm\infty$	DNE	Usually undefined

Common Misconception Check

Myth 1: "The limit at a hole does not exist because the curve is broken."
- Truth: A removable hole leaves both arms converging on the same height. The limit exists at every hole of that kind, which is why the topic is called removable.
Myth 2: "A dot above the hole means the limit is the dot's height."
- Truth: The dot reports the function value. The limit is whatever the arms converge on, and the exam draws these apart precisely to test the distinction.
Myth 3: "If the graph is not defined to the left of a point, the limit does not exist."
- Truth: At an endpoint of the domain only one arm exists, and the one-sided limit is the whole story. Existence then rests on that single side.

Topic-Specific Study Method

The One-Input Sweep

Borrowed idea: Why does the same student who reads a hole correctly then misread a jump two inches to its right? Because they read the whole picture at once and let the most dramatic feature set the tone. Read the graph once per input instead, circling each marked input and answering only the feature inside the circle. One circle, one verdict.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
Two marks, two features	A hole on the left, a jump on the right	Which input does this particular question name?	Answer only the named input, then move to the next question
Statement matching	Four sentences, one graph	Which sentence survives every feature?	Test each sentence at one specific input, not in general
Endpoint item	A curve starting at the left edge	What can be said when only one arm exists?	Read the single arm and call it a one-sided limit
Dot-versus-hole item	Both a hollow circle and a filled dot on one vertical line	Which height is the limit and which is the value?	Arms for the limit, dot for the value, both reported

Executable Action Steps: ① Number the marked inputs before reading anything. Write $a_1, a_2, a_3$ along the axis and put a small tick beside each. Numbering forces you to treat them as separate questions. ② Say the two arm heights out loud, in order. "Left arm reaches three, right arm starts at two." The spoken pair is harder to fudge than a glance. ③ Write DNE with a reason attached. A bare DNE is worth partial credit at best; "DNE because the one-sided limits are three and two" is worth full credit.

Training & Error Diagnosis

Standard Execution:

Left arm: as $x \to 2^-$ , $y = x + 1 \to 3$ .

Right arm: as $x \to 2^+$ , $y = 2x - 1 \to 3$ .

The two arms agree at $3$ , so $\lim_{x \to 2} f(x) = 3$ .

The dot also sits at $3$ , so this graph happens to be continuous at $x = 2$ , even though the rule changes there.

Solution: The two arms converge on $4$ , so the limit is $4$ . The dot reports $f(5) = 1$ . Both answers are correct and they disagree, which is allowed.

2. Endpoint Item. A function is defined only for $x \ge 0$ , and its graph rises to height $2$ as $x \to 0^+$ . What is $\lim_{x \to 0} f(x)$ ?

Solution: Only the right arm exists, and it reaches $2$ . The limit is $2$ , read as the one-sided limit from within the domain. There is nothing to compare on the left.

Common Traps & Failure Signals:

The drama effect (Signal: your answer describes the asymptote when the question names a hole): Visual weight is not the question. Read the input the stem names.
The dot override (Signal: your limit equals the filled dot's height while the arms disagree): The dot is a value, not an approach. If the arms say $4$ and the dot says $1$ , the limit is $4$ .
The averaged jump (Signal: your limit sits between the two arm heights): A two-sided limit requires equality. Halfway is not a height the arms reach.

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

Feature Blending: You merged three questions into one impression.
- Dot Deference: Because the filled dot was drawn darker and larger than the hollow circle right beside it, you treated it as the authoritative answer and reported a function value where the question had asked for an approach height.
- Unstated Verdict: You wrote DNE without naming the two one-sided heights that produced it.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No scored FRQ part in any released BC exam is a pure graph-reading limit item; graphical stems that do appear are read for derivatives or accumulations
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Multi-feature figures where several limits are requested from one picture
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Reading a Removable Hole

The graph of $f$ has a hollow circle at $(3, 6)$ approached by both arms, and a filled dot at $(3, 2)$ . What is $\lim_{x \to 3} f(x)$ ?

MCQ 2

Reading a Jump

The graph of $g$ approaches height $5$ from the left at $x = 1$ and height $-2$ from the right at $x = 1$ . What is $\lim_{x \to 1} g(x)$ ?

MCQ 3

Endpoint Reading

The graph of $h$ is drawn only for $x \ge 2$ , and as $x \to 2^+$ the curve rises to height $7$ . Which statement is true?

FRQ 4

A Piecewise Graph Read at Three Inputs

Let $f$ be defined by $f(x) = \begin{cases} 2 - x, & x < 1 \cr 4, & x = 1 \cr x + 3, & 1 < x < 4 \cr 10 - x, & x \ge 4 \end{cases}$

(a) Find $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$ . Does $\lim_{x \to 1} f(x)$ exist? Justify your answer.

(b) Find $\lim_{x \to 4^-} f(x)$ and $\lim_{x \to 4^+} f(x)$ . Does $\lim_{x \to 4} f(x)$ exist? Justify your answer.

(c) At which of $x = 1$ and $x = 4$ does the limit fail, and at which does the function value differ from the limit? State each answer separately.

Cheatsheet

Core Reference

	What to Look At	What Decides the Answer
Target input	The number named in the question	Not the most eye-catching feature
Left-hand limit	The curve arriving from smaller inputs	Height reached by that arm
Right-hand limit	The curve arriving from larger inputs	Height reached by that arm
Two-sided limit	Both arms together	Equality of two finite heights
Function value	The filled dot on the vertical line	Its height, and nothing else
Hole	A hollow circle where the arms meet	The hole's height is the limit
Jump	Two arms ending at different heights	DNE, with both heights quoted
Asymptote	A curve hugging a vertical line	DNE, with the direction of divergence

Sentences That Score

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