Estimating limit values from graphs

AP Calculus BC· AP Calculus BC CED — Limits and Continuity· 14 min read

1. One-Sided Limits from Graphs★☆☆☆☆⏱ 3 min

📘 Definition

One-Sided Limit

$lim_{x \to a^{-}} f (x)$ (left-hand), $lim_{x \to a^{+}} f (x)$ (right-hand)

A one-sided limit describes the $y$ -value a function approaches as $x$ approaches $a$ from only one direction (left for $x < a$ , right for $x > a$ ), regardless of the actual function value at $x = a$ .

To estimate a one-sided limit from a graph, trace the graph from the direction of the approach, and read the $y$ -value the graph approaches as it nears $x = a$ . Open circles, closed circles, and undefined points do not change the limit—only the trend of the graph near $a$ from the given side matters.

📐 Worked Example

The graph of $f (x)$ has a jump discontinuity at $x = 4$ : for $x < 4$ , the graph approaches an open circle at $(4, 1)$ ; for $x > 4$ , the graph approaches an open circle at $(4, - 5)$ ; and $f (4) = 1$ is marked by a closed circle at $(4, 1)$ . Estimate $lim_{x \to 4^{-}} f (x)$ and $lim_{x \to 4^{+}} f (x)$ .

1
Approach $x = 4$ from the direction of $x < 4$ , tracing the graph toward $x = 4$ .
2
The graph approaches a $y$ -value of $1$ at the open circle $(4, 1)$ , so the left-hand limit equals $1$ .
3
The closed circle at $(4, 1)$ matches the approached value, but the result would be identical even if the closed circle were placed at a different $y$ -value.
4
For the right-hand limit, approach $x = 4$ from the direction of $x > 4$ , tracing the graph toward $x = 4$ .
5
The graph approaches a $y$ -value of $- 5$ at the open circle $(4, - 5)$ , so the right-hand limit equals $- 5$ .
6
Final estimates:
7
$\lim_{x \to 4^-} f(x) = 1 \quad \text{and} \quad \lim_{x \to 4^+} f(x) = -5$

Exam tip:

On AP FRQs, always explicitly mention 'approaching from the left/right' in your justification for a one-sided limit to earn full points.

2. Two-Sided Limits from Graphs★★☆☆☆⏱ 4 min

📘 Definition

Two-Sided Limit

$lim_{x \to a} f (x) = L$

A two-sided limit exists and equals finite value $L$ if and only if both the left-hand and right-hand limits as $x$ approaches $a$ exist and are equal to $L$ . The value or existence of $f (a)$ does not affect the existence or value of the two-sided limit.

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

To estimate a two-sided limit from a graph, always first estimate both one-sided limits, then check for agreement. If they match, that matching value is your two-sided limit. If they do not match, the two-sided limit does not exist (DNE). Limits can exist at points where $f (a)$ is undefined (a hole/point discontinuity) or where $f (a)$ is defined at a different $y$ -value than the limit.

📐 Worked Example

The graph of $g (x) = \frac{x ^{2} - 9}{x - 3}$ matches the line $y = x + 3$ everywhere except at $x = 3$ , where it has a hole at $(3, 6)$ and is undefined. Estimate $lim_{x \to 3} g (x)$ .

1
Find the left-hand limit: approaching $x = 3$ from the left ( $x < 3$ ), the graph approaches $y = 6$ , so $lim_{x \to 3^{-}} g (x) = 6$ .
2
Next find the right-hand limit: approaching $x = 3$ from the right ( $x > 3$ ), the graph also approaches $y = 6$ , so $lim_{x \to 3^{+}} g (x) = 6$ .
3
Both one-sided limits are equal to $6$ , even though $g (3)$ is undefined.
4
By the two-sided limit existence rule:
5
$\lim_{x \to 3} g(x) = 6$

Exam tip:

If an MCQ option says a limit does not exist, double-check that the one-sided limits actually do not match. Many students pick DNE incorrectly when limits match but the function is undefined at the point.

3. Infinite Limits and Limits at Infinity★★☆☆☆⏱ 4 min

Two additional common limit types estimated from graphs are infinite limits (near vertical asymptotes) and limits at infinity (end behavior).

📘 Definition

Infinite Limit

$lim_{x \to a} f (x) = \pm \infty$

Describes the unbounded behavior of $f (x)$ as $x$ approaches a vertical asymptote at $x = a$ . Even though we write this with $\pm \infty$ , an infinite limit does not exist as a finite real number.

📘 Definition

Limit at Infinity

$lim_{x \to \pm \infty} f (x) = L$

Describes the end behavior of $f (x)$ as $x$ grows without bound in the positive or negative direction. If the graph approaches a horizontal asymptote $y = L$ , the limit equals $L$ . If it grows without bound, the limit is infinite (DNE as finite).

📐 Worked Example

The graph of $h (x) = \frac{2 x}{x - 4}$ has a vertical asymptote at $x = 4$ and a horizontal asymptote at $y = 2$ . Estimate (a) $lim_{x \to 4^{-}} h (x)$ , (b) $lim_{x \to 4^{+}} h (x)$ , (c) $lim_{x \to \infty} h (x)$ .

1
(a) Approaching $x = 4$ from the left ( $x < 4$ ), the graph decreases without bound toward the bottom of the coordinate plane, so:
2
$\lim_{x \to 4^-} h(x) = -\infty$
3
(b) Approaching $x = 4$ from the right ( $x > 4$ ), the graph increases without bound toward the top of the plane, so:
4
$\lim_{x \to 4^+} h(x) = \infty$
5
(c) As $x$ grows without bound to the right, the graph approaches the horizontal asymptote $y = 2$ , so:
6
$\lim_{x \to \infty} h(x) = 2$

Exam tip:

If an AP question asks 'does the limit exist' for an infinite limit, you must answer no. Writing $lim_{x \to a} f (x) = \infty$ describes behavior, but it does not mean the limit exists as a finite value.

4. AP-Style Practice Check★★☆☆☆⏱ 3 min

✓ Quick check

Test your understanding with this AP-style multiple choice question:

The graph of $f (x)$ has the following features at $x = - 1$ : For $x < - 1$ , the graph approaches an open circle at $(- 1, 4)$ as $x \to - 1^{-}$ ; For $x > - 1$ , the graph approaches an open circle at $(- 1, - 2)$ as $x \to - 1^{+}$ ; The function is defined at $x = - 1$ with $f (- 1) = 4$ , marked by a closed circle at $(- 1, 4)$ . What is the value of $lim_{x \to - 1} f (x)$ ?
- A) $4$
- B) $- 2$
- C) The limit does not exist
- D) $0$
Reveal answer
C) The limit does not exist —
A two-sided limit exists only if left and right one-sided limits are equal. Here, left-hand limit is $4$ and right-hand limit is $- 2$ , which are not equal. The value of $f (- 1)$ does not affect the existence of the limit.

📐 Worked Example

The graph of $g (x)$ has the following key features: Vertical asymptote at $x = 1$ ; Hole at $x = 3$ at $(3, 5)$ , with $g (3) = 1$ marked by a closed circle at $(3, 1)$ ; Horizontal asymptote at $y = 2$ as $x \to \infty$ . (a) Estimate $lim_{x \to 3} g (x)$ . Justify your answer. (b) As $x \to 1^{-}$ , $g (x)$ increases without bound, and as $x \to 1^{+}$ , $g (x)$ decreases without bound. Write the one-sided limits using correct notation, and state whether $lim_{x \to 1} g (x)$ exists as a finite number. (c) Estimate $lim_{x \to \infty} g (x)$ and explain what this means in terms of the graph's end behavior.

1
(a) As $x$ approaches 3 from both the left and right, the graph approaches the hole at $(3, 5)$ , so both one-sided limits equal 5. Since the one-sided limits are equal:
2
$\lim_{x \to 3} g(x) = 5$
3
The value $g (3) = 1$ does not change the limit, because limits depend on behavior near $x = 3$ , not at $x = 3$ .
4
(b) The one-sided limits are written as:
5
$\lim_{x \to 1^-} g(x) = \infty \quad \text{and} \quad \lim_{x \to 1^+} g(x) = -\infty$
6
The one-sided limits do not agree and neither is finite, so $lim_{x \to 1} g (x)$ does not exist as a finite number.
7
(c) As $x$ grows without bound to the right, the graph approaches the horizontal asymptote at $y = 2$ , so:
8
$\lim_{x \to \infty} g(x) = 2$
9
This means that as $x$ gets larger and larger, the value of $g (x)$ gets arbitrarily close to 2.

5. Common Pitfalls

Wrong move:

For a graph with a hole at $(a, L)$ and $f (a) = M \neq = L$ , you state $lim_{x \to a} f (x) = M$ .

Why:

Students confuse the actual value of the function at $a$ with the value the function approaches near $a$ .

Correct move:

Always ignore the value of $f (a)$ (marked by the closed circle) when calculating a limit; only use the $y$ -value of the open circle or the trend of the graph near $x = a$ .

Wrong move:

When one-sided limits are both equal to $2$ , you state the limit does not exist because $f (a)$ is undefined.

Why:

Students incorrectly assume a limit can't exist if the function doesn't exist at the point.

Correct move:

Existence of a limit at $a$ depends only on the agreement of one-sided limits near $a$ , not on whether $f (a)$ is defined. If one-sided limits agree, the limit exists regardless of $f (a)$ .

Wrong move:

For a jump discontinuity with left limit $3$ and right limit $3$ , but a closed circle at $4$ on one side, you conclude the limit DNE.

Why:

Students confuse the position of the closed circle (the function value) with the limit of the graph's trend.

Correct move:

Check only the one-sided limits from the graph trend; if both approach $3$ , the limit is $3$ regardless of where the closed circle is placed.

Wrong move:

You write $lim_{x \to a} f (x) = \infty$ and then claim the limit exists.

Why:

Students think labeling the behavior as infinity means the limit exists.

Correct move:

On the AP exam, if asked whether the limit exists, you must state that infinite limits do not exist as finite real numbers, even if you can describe their behavior with $\pm \infty$ .

Wrong move:

For a limit as $x \to \infty$ , you approximate the value from the largest visible $x$ on the graph and ignore the horizontal asymptote trend.

Why:

Students use the nearest visible point instead of following the end behavior.

Correct move:

For limits at infinity, always follow the trend of the graph to the far left or far right to find the horizontal asymptote, don't just read the value at the largest visible $x$ .

6. Quick Reference Cheatsheet

Category	Rule/Notation	Key Notes
Left-hand limit	$lim_{x \to a^{-}} f (x)$	Value $f (x)$ approaches as $x$ approaches $a$ from $x < a$ ; only depends on behavior left of $a$
Right-hand limit	$lim_{x \to a^{+}} f (x)$	Value $f (x)$ approaches as $x$ approaches $a$ from $x > a$ ; only depends on behavior right of $a$
Two-sided limit existence	$lim_{x \to a} f (x) = L ⟺ lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = L$	Limit exists and equals $L$ if and only if one-sided limits match; $f (a)$ does not need to be defined at $a$
Infinite limit (vertical asymptote)	$lim_{x \to a} f (x) = \pm \infty$	Describes unbounded growth near $x = a$ ; does not exist as a finite value
Limit at infinity	$lim_{x \to \pm \infty} f (x) = L$	Equals the $y$ -value of the horizontal asymptote the graph approaches as $x \to \pm \infty$
Hole (point discontinuity)	$lim_{x \to a} f (x) = y$ -coordinate of hole	$f (a)$ is undefined at the hole, but the limit still equals the $y$ -coordinate
Jump discontinuity	Two-sided limit DNE	One-sided limits are finite but unequal, so two-sided limit does not exist
Function value vs limit	$f (a) \neq = L ⟹$ no conclusion for the limit	Limit is independent of the actual function value at $a$

What's Next

Estimating limit values from graphs builds the core intuitive understanding of limits that all subsequent calculus work relies on. Immediately after this topic, you will learn algebraic techniques for calculating limits, and the graphical intuition you gain here will help you catch algebraic errors and interpret results when functions are only given graphically, a common AP exam scenario. This topic is also a prerequisite for classifying discontinuities and testing for continuity, the next major topic in Unit 1. Longer term, this understanding of limit behavior from graphs supports work on the definition of the derivative, improper integrals, and end behavior of rational functions later in the course. Without mastering this skill, you will struggle to connect abstract limit definitions to concrete function behavior on exam problems.