Teacher Guide: Evaluating Limits Graphically

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

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10 questions on Limits. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Read $\lim_{x \to a} f(x)$ from a graph by following the curve from both sides
Tell apart the limit at $a$ and the function value $f(a)$ using open circles and filled dots
Recognize that a limit can exist where $f(a)$ is undefined or different from the limit
Decide from a graph when a limit does not exist because the two sides approach different heights
Read several limits and function values from one piecewise graph

Prerequisites

• Introduction to Limits: the idea of a value a function approaches
• Limit Notation: reading $\lim_{x \to a} f(x) = L$
• Reading points and heights on a coordinate graph
• Basic piecewise functions and the meaning of open and filled circles

Discussion Starters

1. If you erased one single point from a graph, could that change the limit there? Why or why not?
2. How can a function have a value at $x = a$ but no limit there?
3. Why do we need to look at both sides of $x = a$ before we name the limit?
4. Where in real life does a graph jump suddenly? What would the limit at the jump mean?

Common Misconceptions

The limit at $a$ is always $f(a)$

Remediation: Show a line with a hole at $(2, 3)$ and a dot at $(2, 5)$ . Ask where the line is heading, then where the dot is. Two different questions, two different answers.

A hole means the limit does not exist

Remediation: Have students walk two fingers toward the hole from both sides. They meet at the hole, so the limit exists. Only a jump, where the fingers end at different heights, makes the limit fail.

The answer to a limit is the number $x$ approaches

Remediation: Insist on reading answers on the $y$ -axis. Have students say 'as $x$ goes to $2$ , the height goes to ...' before writing anything.

Differentiation Ideas

For Struggling Students:

• Start with graphs that have only one special point and grid-aligned heights
• Use two colored markers to trace the left and right sides before naming the limit
• Keep a two-column table for each point: limit (follow the curve) and value (filled dot)

For On-Level Students:

• Read limits and function values at several points of one piecewise graph
• Sort cards with graphs into: limit equals value, limit differs from value, limit does not exist
• Sketch a graph that matches given limits and values

For Advanced Students:

• Design a function whose limit exists at $x = 1$ but differs from $f(1)$ , and write its piecewise formula
• Find a constant that makes two pieces meet so that the limit exists
• Preview one-sided limit notation by describing the left and right heights at a jump

Standards Alignment

HSF-IF.B.4 (CCSS.MATH.CONTENT.HSF.IF.B.4)
Interpret key features of graphs of functions and sketch graphs showing key features
HSF-IF.C.7b (CCSS.MATH.CONTENT.HSF.IF.C.7.B)
Graph piecewise-defined functions, including step functions
AP Calculus LIM-1.C (AP.CALC.LIM-1.C)
Estimate limits of functions from graphs

Lesson Resources

visualHole and Dot Graph
Interactive graph of $y = x + 1$ with a hole at $(2, 3)$ and a filled dot at $(2, 5)$ to compare the limit with $f(2)$
visualPiecewise Graph with Limits
Interactive piecewise graph with a jump and a hole that lists both sides and the function value at each break
activityFinger Walk
Students trace printed graphs with two fingers from both sides toward each marked $x$ and record the heights they reach

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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 $2$ .

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

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

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).

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

Teacher Guide: Evaluating Limits Graphically

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

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