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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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Class quiz

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

For Teachers

Learning Objectives
  • Read lim⁡x→af(x)\lim_{x \to a} f(x) from a graph by following the curve from both sides
  • Tell apart the limit at aa and the function value f(a)f(a) using open circles and filled dots
  • Recognize that a limit can exist where f(a)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→af(x)=L\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=ax = a but no limit there?
  • 3. Why do we need to look at both sides of x=ax = 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 aa is always f(a)f(a)

Remediation: Show a line with a hole at (2,3)(2, 3) and a dot at (2,5)(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 xx approaches

Remediation: Insist on reading answers on the yy-axis. Have students say 'as xx goes to 22, 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=1x = 1 but differs from f(1)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+1y = x + 1 with a hole at (2,3)(2, 3) and a filled dot at (2,5)(2, 5) to compare the limit with f(2)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 xx 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→af(x)\lim_{x \to a} f(x) directly from the graph of ff. Put a finger on the curve on each side of x=ax = a, slide toward x=ax = a, and watch which height (yy-value) the curve is heading for.
  • If both sides head for the same height LL, then lim⁡x→af(x)=L\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→af(x)\lim_{x \to a} f(x) asks where the curve is heading near x=ax = a.
  • f(a)f(a) asks where the filled dot at x=ax = 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=ax = a, so a hole, or a dot sitting somewhere else, does not change the limit.
lim⁡x→af(x)=Lwhen both sides of the graph approach the height L\lim_{x \to a} f(x) = L \quad \text{when both sides of the graph approach the height } L

Worked Examples

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

Follow the curve from the left

As xx goes 1.9,1.99,1.9991.9, 1.99, 1.999, the line rises toward the open circle → Height approaches 33
2

Follow the curve from the right

As xx goes 2.1,2.01,2.0012.1, 2.01, 2.001, the line falls toward the same open circle → Height approaches 33
3

Compare the two sides

Both sides head for the same height → lim⁡x→2f(x)=3\lim_{x \to 2} f(x) = 3
4

Find the filled dot at x=2x = 2

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

Common Mistakes

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

Why it's wrong: The limit describes where the curve is heading near x=2x = 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 33, so lim⁡x→2f(x)=3\lim_{x \to 2} f(x) = 3, while f(2)=5f(2) = 5.

Saying the limit does not exist because there is a hole

Why it's wrong: A hole only means f(a)f(a) is missing. The limit never looks at x=ax = 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)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 xx-value: writing lim⁡x→2f(x)=2\lim_{x \to 2} f(x) = 2

Why it's wrong: The 22 under the limit sign is where xx goes. The limit itself is a height, a yy-value.

Correct: Read the answer on the yy-axis: the height the curve approaches as xx gets close to 22.

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 aa exactly when the limit exists and equals f(a)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)C(w) has a filled dot at (2,5)(2, 5) and an open circle at (2,8)(2, 8).

How big is the jump at w=2w = 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)T(t) in °C, with tt in hours. Around t=3t = 3 the curve approaches 1818 °C from both sides, but a glitch stored the single value T(3)=0T(3) = 0.

What is lim⁡t→3T(t)\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)=x2−4x−2f(x) = \frac{x^2 - 4}{x - 2} the graph looks like y=x+2y = x + 2 with a hole at (2,4)(2, 4), so lim⁡x→2f(x)=4\lim_{x \to 2} f(x) = 4 while f(2)f(2) is undefined.

3Try It Yourself

You graph f(x)=x2−9x−3f(x) = \frac{x^2 - 9}{x - 3}. Tracing shows ERROR at x=3x = 3, and the graph looks like the line y=x+3y = x + 3 with a hole.

What is lim⁡x→3f(x)\lim_{x \to 3} f(x)?

Step 1: Write the mathematical expression

The height of the hole:

Key Takeaways

  • 1To read lim⁡x→af(x)\lim_{x \to a} f(x) from a graph, follow the curve from both sides toward x=ax = a and read the height it approaches
  • 2The limit is a yy-value (a height), not the xx-value aa
  • 3An open circle (hole) marks a missing point, a filled dot marks the actual value f(a)f(a)
  • 4The limit ignores the point at x=ax = 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)f(a) is
  • 6Where the graph has no break, the limit simply equals f(a)f(a)

Frequently Asked Questions

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

Yes. If the graph has a hole at x=ax = a and both sides approach the height of the hole, the limit equals that height. For example x2−4x−2\frac{x^2 - 4}{x - 2} is undefined at x=2x = 2, but its limit there is 44.

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

No. The dot only gives f(a)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→af(x)\lim_{x \to a} f(x) does not exist (often shortened to DNE) and give the reason, for example: the left side approaches 22 and the right side approaches 44.

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 LL that the graph of ff approaches from both sides as xx approaches aa, written lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
Function value f(a)f(a)
The actual height of the graph at x=ax = a, shown by a filled dot or by the curve passing through x=ax = 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→af(x)=L\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

left height≠right height  ⇒  lim⁡x→af(x) 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→af(x) and f(a) can differ\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

More in This Topic

Introduction to Limits

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Limit Notation

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Evaluating Limits from Tables

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Limits by Direct Substitution

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