Estimating limit values from graphs — AP Calculus AB Study Guide

For: AP Calculus AB candidates sitting AP Calculus AB.

Covers: One-sided limits, two-sided limits, infinite limits, limits at infinity, distinguishing limit values from function values, and estimating all limit types from labeled function graphs.

You should already know: Basic limit notation for one-sided and two-sided limits. How to read coordinate values from a function graph. Definitions of common function discontinuity types.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Calculus AB style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Estimating limit values from graphs?

Estimating limit values from graphs is the process of using the visual behavior of a function’s curve near an input value $x=a$ to determine what output value the function approaches as $x$ gets arbitrarily close to $a$, regardless of the actual value of $f(a)$ itself. Per the AP Calculus AB Course and Exam Description (CED), this is Topic 1.2 in Unit 1: Limits and Continuity, which accounts for 10-12% of the total AP exam score. This topic appears in both multiple-choice (MCQ) and free-response (FRQ) sections, almost always as a standalone question or a lead-in for later continuity or derivative problems.

This topic builds the intuitive foundation for all limit work, teaching you to separate the core idea of a limit (what the function approaches near a point) from the function’s actual value at that point. On the AP exam, you will typically be given a labeled graph with no explicit function formula, and asked to calculate one or more limit values or compare limits to function values.

Quick Worked Example

Problem: A graph of $f(x)$ has $f(2)=5$, and the curve approaches $y=3$ as $x$ approaches 2 from both sides. What is ${\lim }_{x\to 2}f(x)$? Solution: The limit asks for the value the function approaches near $x=2$, so ${\lim }_{x\to 2}f(x)=3$, not 5.

2. One-Sided Limits from Graphs

A one-sided limit describes the output value a function approaches as we approach $x=a$ from only one side: the left (all $x<a$, moving right toward $a$) or the right (all $x>a$, moving left toward $a$). Standard notation follows clear conventions: $$\underset{x\to a^{-}}{\lim }f(x)=\text{Left-hand limit (approaching from left of }a\text{)}$$ $$\underset{x\to a^{+}}{\lim }f(x)=\text{Right-hand limit (approaching from right of }a\text{)}$$ To estimate a one-sided limit from a graph, you trace the curve along the specified side of $x=a$, and find the $y$-coordinate that the curve approaches as you get arbitrarily close to the vertical line $x=a$. Critically, the function’s value at $x=a$, or its behavior on the opposite side of $a$, does not affect the one-sided limit at all. This makes one-sided limits the building block for all other limit calculations from graphs.

Worked Example

Problem: The graph of $f(x)$ has a jump discontinuity at $x=2$. For $x<2$, the curve approaches an open circle at $(2,3)$, and for $x>2$, the curve approaches an open circle at $(2,-1)$. The function is defined as $f(2)=3$. Estimate ${\lim }_{x\to 2^{-}}f(x)$ and ${\lim }_{x\to 2^{+}}f(x)$.

1. For ${\lim }_{x\to 2^{-}}f(x)$, we only consider behavior for $x<2$ approaching 2 from the left.
2. Tracing the curve from the left, it approaches the open circle at $y=3$ at $x=2$, so ${\lim }_{x\to 2^{-}}f(x)=3$.
3. For ${\lim }_{x\to 2^{+}}f(x)$, we only consider behavior for $x>2$ approaching 2 from the right.
4. Tracing the curve from the right, it approaches the open circle at $y=-1$ at $x=2$, so ${\lim }_{x\to 2^{+}}f(x)=-1$.

Exam tip: On AP MCQ questions, always double-check the exponent sign: $a^{-}$ means left (less than $a$) and $a^{+}$ means right (greater than $a$) — mixing these up is the most common careless error on one-sided limit questions.

3. Two-Sided Limits from Graphs

A two-sided limit ${\lim }_{x\to a}f(x)$ is the output value that $f(x)$ approaches as $x$ approaches $a$ from both the left and the right. The core existence rule for two-sided limits is:

A two-sided limit exists if and only if both the left-hand and right-hand one-sided limits exist and are equal. If ${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=L$, then ${\lim }_{x\to a}f(x)=L$. If the one-sided limits are not equal, the two-sided limit does not exist (DNE).

When estimating from a graph, the process is always: first find both one-sided limits, check for equality, then state your result. The most common tested scenario for two-sided limits is a removable discontinuity (a hole) at $x=a$: even if $f(a)$ is undefined or defined at a different $y$-value, both one-sided limits approach the same value, so the two-sided limit equals that $y$-value regardless of $f(a)$.

Worked Example

Problem: The graph of $g(x)$ has an open circle at $(4,-2)$ and a closed defined point at $(4,5)$. As $x$ approaches 4 from both the left and the right, $g(x)$ approaches the open circle at $(4,-2)$. Find ${\lim }_{x\to 4}g(x)$.

1. First calculate the left-hand limit: tracing from the left of 4, $g(x)$ approaches $y=-2$, so ${\lim }_{x\to 4^{-}}g(x)=-2$.
2. Next calculate the right-hand limit: tracing from the right of 4, $g(x)$ also approaches $y=-2$, so ${\lim }_{x\to 4^{+}}g(x)=-2$.
3. Check if the one-sided limits are equal: both are equal to $-2$, so the two-sided limit exists.
4. The value of $g(4)=5$ is irrelevant to the limit calculation, because the limit only describes behavior near $x=4$, not at $x=4$. So ${\lim }_{x\to 4}g(x)=-2$.

Exam tip: The AP exam almost always tests the distinction between the two-sided limit and the function value at $x=a$ by placing the closed point at a different $y$-coordinate than the approached open circle. Never automatically set the limit equal to $f(a)$.

4. Infinite Limits and Limits at Infinity from Graphs

Beyond limits at finite $x=a$, we also estimate two special limit types from graphs: infinite limits (where the function approaches $\pm \infty$ as $x$ approaches a finite $a$) and limits at infinity (where $x$ approaches $\pm \infty$, and we estimate what $y$-value the function approaches).

An infinite limit occurs at a vertical asymptote $x=a$: as $x$ approaches $a$ from one or both sides, the function grows without bound toward positive or negative infinity. If both sides approach the same signed infinity, we write ${\lim }_{x\to a}f(x)=+\infty$ or $-\infty$. If the two sides approach opposite infinities, the two-sided limit DNE.

A limit at infinity describes end behavior of the function: as $x$ grows arbitrarily large positive ($x\to +\infty$) or arbitrarily large negative ($x\to -\infty$), what $y$-value does the function approach? If the graph approaches a horizontal line $y=L$, that $L$ is the limit at infinity; if the function grows without bound, the limit is $\pm \infty$.

Worked Example

Problem: The graph of $h(x)$ has a vertical asymptote at $x=2$. As $x\to 2^{-}$, $h(x)$ approaches $+\infty$, and as $x\to 2^{+}$, $h(x)$ approaches $-\infty$. As $x\to +\infty$, the graph approaches the line $y=2$, a horizontal asymptote. Estimate (a) ${\lim }_{x\to 2}h(x)$ and (b) ${\lim }_{x\to +\infty }h(x)$.

1. For part (a), we first get the one-sided limits: ${\lim }_{x\to 2^{-}}h(x)=+\infty$ and ${\lim }_{x\to 2^{+}}h(x)=-\infty$.
2. The one-sided limits are not equal, so the two-sided limit ${\lim }_{x\to 2}h(x)$ does not exist.
3. For part (b), we look at end behavior as $x$ grows large positive: the graph approaches the horizontal asymptote $y=2$, getting arbitrarily close to $y=2$ as $x$ increases.
4. Therefore, ${\lim }_{x\to +\infty }h(x)=2$.

Exam tip: If the function approaches an infinite limit from both sides, you must state the signed infinity (e.g. ${\lim }_{x\to a}f(x)=+\infty$) instead of just writing DNE to earn full credit on FRQ questions.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Stating ${\lim }_{x\to a}f(x)=f(a)$ because the closed point is at $(a,f(a))$ even though both sides of the graph approach a different $y$-value. Why: Students confuse the definition of a function value at a point with the definition of a limit, which describes behavior near the point, not at the point. Correct move: Always ignore the closed point's $y$-coordinate when estimating the limit, unless you confirm the function is continuous at $a$.
• Wrong move: Mixing up left-hand and right-hand limit notation, reporting the right-hand limit when asked for the left-hand limit, or vice versa. Why: Students associate the minus sign with negative $y$ instead of approaching from values less than $a$, and plus with positive $y$ instead of values greater than $a$. Correct move: Memorize the mnemonic: $a^{-}$ = left (less than $a$), $a^{+}$ = right (greater than $a$), and write it on scratch paper if needed.
• Wrong move: Stating that the two-sided limit exists if only one one-sided limit exists, even if the other does not exist or they are not equal. Why: Students forget the "both exist and are equal" requirement for two-sided limits. Correct move: Always calculate both one-sided limits first, check for equality, then conclude if the two-sided limit exists.
• Wrong move: Estimating the limit as $x\to a$ by reporting the $x$-coordinate $a$ instead of the approached $y$-coordinate. Why: Students mix up input and output when reading questions. Correct move: Remember that ${\lim }_{x\to a}f(x)$ always asks for an output ($y$) value, never an input ($x$) value.
• Wrong move: Stating that a limit at infinity does not exist because the function never actually reaches the horizontal asymptote or oscillates around it while getting closer. Why: Students think the function has to reach the limit value for the limit to exist. Correct move: If the function gets arbitrarily close to $y=L$ as $x\to \infty$, the limit is $L$ regardless of whether it ever equals $L$.
• Wrong move: Reporting "does not exist" for an infinite limit that approaches the same signed infinity from both sides. Why: Students know infinite limits are not finite, so they generalize to DNE, missing context. Correct move: If both sides approach the same signed infinity, write the limit as $+\infty$ or $-\infty$; only write DNE if the two sides differ.

6. Practice Questions (AP Calculus AB Style)

Question 1 (Multiple Choice)

The graph of a function $f$ has the following behavior: At $x=-1$, $f$ has a jump discontinuity. As $x$ approaches $-1$ from the left, $f(x)$ approaches $2$, and as $x$ approaches $-1$ from the right, $f(x)$ approaches $-4$. The value of $f(-1)$ is $2$. At $x=3$, $f$ has a removable discontinuity: the graph approaches an open circle at $(3,4)$ from both sides, and the closed point for $f(3)$ is at $(3,1)$. What is the value of ${\lim }_{x\to 3^{-}}f(x)+{\lim }_{x\to -1^{+}}f(x)$?

A) $-5$ B) $0$ C) $3$ D) $6$

Worked Solution: First, we apply the one-sided limit rule. For ${\lim }_{x\to 3^{-}}f(x)$, we look at behavior approaching 3 from the left: the graph approaches the open circle at $y=4$, so this limit equals 4. The closed point value $f(3)=1$ is a distractor, since the limit does not depend on the function's value at the point. Next, ${\lim }_{x\to -1^{+}}f(x)$ is the limit approaching $-1$ from the right, which is given as $-4$. Adding the two limits gives $4+(-4)=0$. The correct answer is B.

Question 2 (Free Response)

The graph of a function $y=k(x)$ has the following key features:

• Vertical asymptote at $x=1$
• As $x\to 1^{-}$, $k(x)$ approaches $3$; as $x\to 1^{+}$, $k(x)$ approaches $+\infty$
• At $x=2$, the graph has a closed point at $(2,0)$ and an open point at $(2,-2)$; the graph approaches the open point from both sides
• As $x\to +\infty$, $k(x)$ approaches the horizontal line $y=-1$

(a) Identify ${\lim }_{x\to 1^{-}}k(x)$ and ${\lim }_{x\to 1^{+}}k(x)$. Does ${\lim }_{x\to 1}k(x)$ exist? Justify your answer. (b) Find ${\lim }_{x\to 2}k(x)$ and $k(2)$. Explain why the limit is not equal to the function value at $x=2$. (c) Find ${\lim }_{x\to +\infty }k(x)$. What does this limit represent in terms of the graph's end behavior?

Worked Solution: (a) From the graph features: ${\lim }_{x\to 1^{-}}k(x)=3$ and ${\lim }_{x\to 1^{+}}k(x)=+\infty$. ${\lim }_{x\to 1}k(x)$ does not exist. Justification: A two-sided limit requires both one-sided limits to exist as finite equal values. The right-hand limit is infinite, so the two-sided limit cannot exist.

(b) ${\lim }_{x\to 2}k(x)=-2$, because the left-hand and right-hand limits both approach $-2$ at $x=2$. $k(2)=0$, the value of the closed point at $x=2$. The limit is not equal to the function value because $k(x)$ has a removable discontinuity at $x=2$: the limit describes behavior near $x=2$, which approaches $-2$, regardless of the defined value at $x=2$ itself.

(c) ${\lim }_{x\to +\infty }k(x)=-1$. This means that as $x$ grows arbitrarily large and positive, the $y$-value of $k(x)$ gets arbitrarily close to $-1$, which corresponds to the horizontal asymptote $y=-1$ for the graph of $k(x)$.

Question 3 (Application / Real-World Style)

A physics lab measures the velocity of a block sliding down a ramp that ends in a collision with a spring at $t=1.2$ seconds after the block starts moving. The function $v(t)$ gives the velocity of the block (in m/s) at time $t$ seconds. The graph of $v(t)$ shows that before $t=1.2$, the block is accelerating, and as $t$ approaches 1.2 from the left, $v(t)$ approaches 4.5 m/s. At $t=1.2$, the block hits the spring, so $v(1.2)=0$, and for all $t>1.2$, $v(t)=0$ as the spring holds the block in place. Estimate ${\lim }_{t\to 1.2^{-}}v(t)$ and ${\lim }_{t\to 1.2}v(t)$, and interpret your results in context.

Worked Solution: First, ${\lim }_{t\to 1.2^{-}}v(t)$ is the velocity the block approaches as it gets to the collision point from earlier times. From the graph, this limit is 4.5 m/s. Next, to find the two-sided limit, we calculate the right-hand limit: for $t>1.2$, $v(t)=0$, so ${\lim }_{t\to 1.2^{+}}v(t)=0$. Since $4.5\ne 0$, the two-sided limit ${\lim }_{t\to 1.2}v(t)$ does not exist. Interpretation: The one-sided limit of 4.5 m/s means that just before the block collided with the spring at 1.2 seconds, its velocity was approaching 4.5 meters per second. The two-sided limit does not exist because the velocity drops instantly to 0 after the collision, so the behavior on either side of 1.2 seconds does not approach the same value.

7. Quick Reference Cheatsheet

8. What's Next

This topic is the intuitive foundation for all future work with limits and continuity in AP Calculus AB. Immediately after mastering estimating limits from graphs, you will move on to estimating limit values from tables, then algebraic techniques for calculating limits exactly. Without the core understanding that a limit describes behavior near a point (not at the point) that you gain from this chapter, you will struggle to apply algebraic limit rules, identify discontinuities, or compute derivatives via the limit definition later in the course. This topic also builds intuition for end behavior and asymptotic analysis that is critical for curve sketching and related rates problems later in the course.

• Estimating limit values from tables
• Algebraic techniques for calculating limits
• Continuity of functions at a point
• The definition of the derivative