If there is a filled dot at $x = a$, does that mean the limit exists?

No. The dot only gives $f(a)$. At a jump the graph has a filled dot, but the two sides approach different heights, so the limit does not exist.

What do I write when a limit does not exist?

Write $\lim_{x \to a} f(x)$ does not exist (often shortened to DNE) and give the reason, for example: the left side approaches $2$ and the right side approaches $4$.

How accurate is reading a limit from a graph?

Only as accurate as the graph. Use grid lines and labelled points, and when the height is not a clear grid value, confirm it with a table of values or with algebra (the next lessons).

Teacher Resources

Evaluating Limits Graphically

Q: Can the limit exist if $f(a)$ is undefined?

Yes. If the graph has a hole at $x = a$ and both sides approach the height of the hole, the limit equals that height. For example $\frac{x^2 - 4}{x - 2}$ is undefined at $x = 2$, but its limit there is $4$.

Read limits straight from a graph, tell the limit apart from the function value, and spot when a limit does not exist.

Advanced25 minLesson

Printable worksheet

Definition

To evaluate a limit graphically, you read

\lim_{x \to a} f(x)

directly from the graph of

f

. Put a finger on the curve on each side of

x = a

, slide toward

x = a

, and watch which height (

y

-value) the curve is heading for.

If both sides head for the same height $L$ , then $\lim_{x \to a} f(x) = L$ .
If the two sides head for different heights, the limit does not exist (DNE).

Limit vs. function value. These are two different questions:

$\lim_{x \to a} f(x)$ asks where the curve is heading near $x = a$ .
$f(a)$ asks where the filled dot at $x = a$ actually sits.

An open circle (a hole) marks a point that is missing from the graph. A filled dot marks the actual value of the function. The limit ignores what happens exactly at

x = a

, so a hole, or a dot sitting somewhere else, does not change the limit.

\lim_{x \to a} f(x) = L \quad \text{when both sides of the graph approach the height } L

Worked Examples

The graph of $f$ is the line $y = x + 1$ with an open circle at $(2, 3)$ and a filled dot at $(2, 5)$ (the first graph in this lesson). Find $\lim_{x \to 2} f(x)$ and $f(2)$ .

Follow the curve from the left

As $x$ goes $1.9, 1.99, 1.999$ , the line rises toward the open circle → Height approaches $3$

Follow the curve from the right

As $x$ goes $2.1, 2.01, 2.001$ , the line falls toward the same open circle → Height approaches $3$

Compare the two sides

Both sides head for the same height → $\lim_{x \to 2} f(x) = 3$

Find the filled dot at $x = 2$

The filled dot sits at $(2, 5)$ , away from the line → $f(2) = 5$

Answer: $\lim_{x \to 2} f(x) = 3$ but $f(2) = 5$ . The limit follows the curve, the function value is the filled dot.

Common Mistakes

Reading the filled dot as the limit: answering $\lim_{x \to 2} f(x) = 5$ because there is a dot at $(2, 5)$

Why it's wrong: The limit describes where the curve is heading near $x = 2$ . A single dot away from the curve does not change where the curve is heading.

Correct: Follow the curve from both sides: it heads for the open circle at height $3$ , so $\lim_{x \to 2} f(x) = 3$ , while $f(2) = 5$ .

Saying the limit does not exist because there is a hole

Why it's wrong: A hole only means $f(a)$ is missing. The limit never looks at $x = a$ itself, only at the values around it.

Correct: If both sides approach the height of the hole, the limit exists and equals that height, even when $f(a)$ is undefined.

At a jump, giving the height of the filled dot (or of one side) as the limit

Why it's wrong: A two-sided limit needs both sides to agree. At a jump, the left side and the right side approach different heights.

Correct: Check both sides. If they approach different heights, write that the limit does not exist, whatever the filled dot shows.

Answering with the $x$ -value: writing $\lim_{x \to 2} f(x) = 2$

Why it's wrong: The $2$ under the limit sign is where $x$ goes. The limit itself is a height, a $y$ -value.

Correct: Read the answer on the $y$ -axis: the height the curve approaches as $x$ gets close to $2$ .

Interactive Visual

Function graph

Tap or hover over the graph to read off values.

At x = 2: from the left and from the right f(x) → 3, so the limit is 3. f(2) = 5.

Function graph

Tap or hover over the graph to read off values.

At x = -1: from the left g(x) → 2, from the right g(x) → 4. The values differ, so the limit does not exist (jump). g(-1) = 2.
At x = 2: from the left and from the right g(x) → 1, so the limit is 1. g(2) is undefined (a hole).

Interactive Sandbox

Interactive Grapher

Tap or hover over the graph to read off values.

Try these examples:

y = (x^2 - 4)/(x - 2)

m=1, b=2

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

18 problems

Problem 1 of 18

Easy

The graph of $f$ is a straight line that runs through an open circle at $(3, 4)$ . There is also a filled dot at $(3, 1)$ . What is $\lim_{x \to 3} f(x)$ ?

Practice Problems: Evaluating Limits Graphically

1. The graph of f is a straight line that runs through an open circle at (3, 4). There is also a filled dot at (3, 1). What is \lim_{x \to 3} f(x)?
2. The graph of g is the parabola y = x^2 with an open circle at (2, 4) and a filled dot at (2, 0). What is \lim_{x \to 2} g(x)?
3. The graph of f is the line y = x + 2 with an open circle at (1, 3) and a filled dot at (1, 6). Find \lim_{x \to 1} f(x) and compare it with f(1).
4. The graph of f is a straight line through an open circle at (3, 4), plus a filled dot at (3, 1). What is f(3)?
5. The graph of g is the parabola y = x² with an open circle at (2, 4) and a filled dot at (2, 0). What is g(2)?
6. The graph of f is the line y = 4 - x, drawn without any gaps or extra dots. Find \lim_{x \to 1} f(x) and check whether it equals f(1).
7. For x < 2 the graph of f follows y = x and ends in an open circle at (2, 2). For x \ge 2 it follows y = 5 - x, starting with a filled dot at (2, 3). What is \lim_{x \to 2} f(x)?
8. The graph of h is the line through (0, -1) and (4, 7), with an open circle at x = 2 and no filled dot there. What is \lim_{x \to 2} h(x)?
9. For x < 2 the graph of f is the line y = x, ending in an open circle at (2, 2). For x \ge 2 it is the line y = x + 3, starting with a filled dot at (2, 5). Does \lim_{x \to 2} f(x) exist, and what is f(2)?
10. Which statement about \lim_{x \to a} f(x) is always true?
11. Near x = -2 the graph of f approaches an open circle at (-2, 3) from both sides, and there is a filled dot at (-2, -1). Compute \lim_{x \to -2} f(x) + f(-2).
12. The graph of g: for x ≤ -1 it is y = x + 3, ending in a filled dot at (-1, 2); for -1 < x < 2 it is y = (x - 1)², starting at an open circle at (-1, 4); for x > 2 it is y = 3 - x. At x = 2 there is an open circle at (2, 1) and no filled dot. Read the limits from the graph.
13. The graph of f(x) = \frac{x^2 - 1}{x + 1} looks like the line y = x - 1 with a hole. Use the graph to find \lim_{x \to -1} f(x).
14. The graph of f has three special places. At x = 1 the left part ends at height 2 and the right part starts at height -1. At x = 3 the curve approaches an open circle at (3, 4) from both sides, with no filled dot. At x = 5 the curve approaches (5, 0) from both sides, but the filled dot is at (5, 3). For which a does \lim_{x \to a} f(x) not exist?
15. The graph of f follows y = 3x + k for x < 2 and y = x^2 for x > 2, with open circles at both ends at x = 2. For which value of k does \lim_{x \to 2} f(x) exist?
16. The graph of q: for x < 2 it is y = x^2, ending in an open circle at (2, 4); there is a filled dot at (2, 1); for 2 < x < 5 it is y = 6 - x, between open circles at (2, 4) and (5, 1); for x \ge 5 it is y = x - 2, starting with a filled dot at (5, 3). Find \lim_{x \to 2} q(x), q(2), and decide whether \lim_{x \to 5} q(x) exists.
17. For x < 3 the graph of f is y = ax, for x > 3 it is y = 12 - x, and there is a filled dot at (3, 0). Find the value of a for which \lim_{x \to 3} f(x) exists.
18. The graph of f: for x \le 0 it is y = -x, ending in a filled dot at (0, 0); for 0 < x < 4 it is y = x + 2, between open circles at (0, 2) and (4, 6); there is a filled dot at (4, 1); for x > 4 it is y = 10 - x, starting at an open circle at (4, 6). Compute \lim_{x \to 4} f(x) + \lim_{x \to 2} f(x) + f(0).

Why It Matters

A graph is often the fastest way to see what a function does near a point:

Calculus starts here: the derivative (slope of a tangent line) is a limit, and you first meet it as a picture.
Real data has gaps and glitches: a sensor graph may have a missing reading or one wrong spike. The limit tells you the true trend around that moment.
Prices and fees jump: shipping costs, taxi fares and tax brackets have graphs with jumps, where the value just before and just after a boundary differ.
Continuity builds on it: later you will call a function continuous at $a$ exactly when the limit exists and equals $f(a)$ , which you can check by eye on a graph.

Reading limits from graphs trains the most important habit in calculus: look at what happens near a point, not only at it.

Real World Applications

Shipping Costs with a Weight Limit

Couriers charge by weight class, so the graph of cost against weight is a step graph with jumps at the class boundaries.

Example:

A courier charges 5 dollars up to 2 kg and 8 dollars above 2 kg. Just below 2 kg the graph sits at 5, just above it sits at 8, so the limit of the cost as the weight approaches 2 kg does not exist.

1Try It Yourself

A courier charges 5 dollars for parcels up to 2 kg (2 kg included) and 8 dollars for parcels from just over 2 kg up to 5 kg. The graph of the cost $C(w)$ has a filled dot at $(2, 5)$ and an open circle at $(2, 8)$ .

How big is the jump at $w = 2$ (the height the right side approaches minus the height the left side approaches)?

Step 1: Write the mathematical expression

Right side height minus left side height:

A Glitch in Sensor Data

Weather stations and fitness trackers plot readings over time. When one reading is lost or wrong, the graph gets a hole or a stray dot, and the limit recovers the real trend.

Example:

A heart rate graph that approaches 120 beats per minute from both sides of one moment, with a single faulty reading of 0 at that moment, still has a limit of 120 there.

2Try It Yourself

A weather station graphs the temperature $T(t)$ in °C, with $t$ in hours. Around $t = 3$ the curve approaches $18$ °C from both sides, but a glitch stored the single value $T(3) = 0$ .

What is $\lim_{t \to 3} T(t)$ ?

Step 1: Write the mathematical expression

The height both sides approach:

Holes on a Graphing Calculator

Graphing software draws $f(x) = \frac{x^2 - 9}{x - 3}$ as what looks like a straight line, but tracing at $x = 3$ gives an error. The graph has a hole there, and its height is the limit.

Example:

For $f(x) = \frac{x^2 - 4}{x - 2}$ the graph looks like $y = x + 2$ with a hole at $(2, 4)$ , so $\lim_{x \to 2} f(x) = 4$ while $f(2)$ is undefined.

3Try It Yourself

You graph $f(x) = \frac{x^2 - 9}{x - 3}$ . Tracing shows ERROR at $x = 3$ , and the graph looks like the line $y = x + 3$ with a hole.

What is $\lim_{x \to 3} f(x)$ ?

Step 1: Write the mathematical expression

The height of the hole:

Key Takeaways

1To read $\lim_{x \to a} f(x)$ from a graph, follow the curve from both sides toward $x = a$ and read the height it approaches
2The limit is a $y$ -value (a height), not the $x$ -value $a$
3An open circle (hole) marks a missing point, a filled dot marks the actual value $f(a)$
4The limit ignores the point at $x = a$ : a hole or a dot somewhere else does not change it
5If the two sides approach different heights (a jump), the limit does not exist, whatever $f(a)$ is
6Where the graph has no break, the limit simply equals $f(a)$

Frequently Asked Questions

Yes. If the graph has a hole at

x = a

and both sides approach the height of the hole, the limit equals that height. For example

\frac{x^2 - 4}{x - 2}

is undefined at

x = 2

, but its limit there is

4

Glossary

Limit: The height $L$ that the graph of $f$ approaches from both sides as $x$ approaches $a$ , written $\lim_{x \to a} f(x) = L$
Function value $f(a)$: The actual height of the graph at $x = a$ , shown by a filled dot or by the curve passing through $x = a$
Open circle (hole): A point drawn as an empty circle: the graph comes close to it, but it is not part of the graph
Filled dot: A solid point that is part of the graph and gives the function value there
Jump: A place where the graph breaks and continues at a different height, so the two sides approach different values
Does not exist (DNE): What we say about a limit when the two sides of the graph do not approach the same height

Formula Card

Reading a limit from a graph

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

Both sides of the graph approach the height $L$ as $x$ approaches $a$

When the limit does not exist

\text{left height} \neq \text{right height} \;\Rightarrow\; \lim_{x \to a} f(x) \text{ does not exist}

If the two sides approach different heights, $\lim_{x \to a} f(x)$ does not exist

Limit vs. value

\lim_{x \to a} f(x) \text{ and } f(a) \text{ can differ}

The limit follows the curve near $a$, the value is the filled dot at $a$. They can be different

Evaluating Limits Graphically

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Function graph

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Shipping Costs with a Weight Limit

A Glitch in Sensor Data

Holes on a Graphing Calculator

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Reading a limit from a graph

When the limit does not exist

Limit vs. value

More in This Topic

Introduction to Limits

Limit Notation

Evaluating Limits from Tables

Limits by Direct Substitution