Understanding Limits

Summary

This chapter introduces the fundamental concept of limits, which forms the cornerstone of calculus. Students will develop an intuitive understanding of what it means for a function to approach a value, learn proper limit notation, and explore one-sided and two-sided limits. The chapter also covers the essential limit laws that allow for algebraic manipulation of limits. After completing this chapter, students will be able to evaluate basic limits and understand when limits exist or fail to exist.

Concepts Covered

This chapter covers the following 15 concepts from the learning graph:

Limit
Limit Notation
Intuitive Limit
One-Sided Limit
Left-Hand Limit
Right-Hand Limit
Two-Sided Limit
Limit Existence
Limit Laws
Sum Rule for Limits
Product Rule for Limits
Quotient Rule for Limits
Constant Multiple Rule
Power Rule for Limits
Direct Substitution

Prerequisites

This chapter builds on concepts from:

Chapter 1: Foundations of Calculus

What Is a Limit?

Welcome to the heart of calculus! If Chapter 1 introduced you to the idea of change, this chapter gives you the precise language to describe it. That language is called limits, and once you master it, you'll have the key that unlocks everything else in calculus.

Delta Moment

"I'm standing at \(x = 2\) on this curve, but I'm curious about what's happening just to my left and right. What value is the function sneaking up on? That's a limit!"

Here's the big idea: a limit describes what value a function approaches as its input gets closer and closer to some target. Notice the word "approaches"—we're not always asking what the function equals at that point. We're asking what it's heading toward.

Think about walking toward a door. The limit is where you're heading, even if you never actually step through. In math terms:

\[\lim_{x \to c} f(x) = L\]

This reads as: "The limit of \(f(x)\) as \(x\) approaches \(c\) equals \(L\)."

Why Limits Matter

Limits solve a problem that plagued mathematicians for centuries: how do you describe something that's changing at a single instant?

Consider your speedometer. It doesn't show your average speed over the last hour—it shows your speed right now. But "right now" is just an instant, with no time elapsed. How can you have speed without time? The answer, as we'll see in later chapters, involves limits.

For now, let's build your intuition with a classic example.

The Intuitive Limit

Consider the function:

\[f(x) = \frac{x^2 - 4}{x - 2}\]

What happens at \(x = 2\)? If you plug in directly, you get \(\frac{0}{0}\)—which is undefined. The function literally has a hole at \(x = 2\).

But here's the interesting question: what is \(f(x)\) approaching as \(x\) gets close to 2?

\(x\)	\(f(x)\)
1.9	3.9
1.99	3.99
1.999	3.999
2.001	4.001
2.01	4.01
2.1	4.1

The pattern is unmistakable. As \(x\) approaches 2, \(f(x)\) approaches 4. We write:

\[\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4\]

The function never equals 4 at \(x = 2\) (there's a hole there!), but 4 is still the limit.

Diagram: Limit Visualization with Hole

Limit Visualization with Hole MicroSim

Type: microsim

Bloom Level: Understand (L2) Bloom Verb: Interpret

Learning Objective: Students will interpret the concept of a limit by observing how function values approach a target as x approaches a specific point, even when the function is undefined at that point.

Visual elements:

Coordinate plane with x-axis from -1 to 5, y-axis from -1 to 8
Graph of \(f(x) = \frac{x^2 - 4}{x - 2}\) which simplifies to \(f(x) = x + 2\) with a hole at \((2, 4)\)
Open circle (hole) at the point \((2, 4)\)
Delta robot character positioned on the curve
Animated point that approaches \(x = 2\) from both sides
Dashed horizontal line at \(y = 4\) (the limit value)
Value display showing current \(x\) and \(f(x)\) values

Interactive controls:

Slider: "Distance from 2" ranging from 0.5 to 0.001
Dropdown: "Approach from" with options "Left," "Right," "Both"
Button: "Animate approach"
Button: "Reset"

Behavior:

As slider moves toward 0, the point on the curve moves toward x = 2
The y-value display updates to show f(x) getting closer to 4
When "Both" is selected, two points animate simultaneously from left and right
Delta tilts to match the slope at the current position

Instructional Rationale: Step-through visualization is appropriate because the Understand/interpret objective requires learners to see the connection between approaching x-values and the resulting y-values converging to the limit. The hole visualization makes explicit that limits describe approach behavior, not actual values.

Data Visibility Requirements:

Stage 1: Show x = 1.5, f(x) = 3.5
Stage 2: Show x = 1.9, f(x) = 3.9
Stage 3: Show x = 1.99, f(x) = 3.99
Stage 4: Show x = 1.999, f(x) = 3.999
Final: Show limit value L = 4 with all approach values visible

Implementation: p5.js with canvas-based controls

Limit Notation

Now that you understand what limits mean intuitively, let's nail down the notation you'll use throughout calculus.

Reading Limit Notation

The expression \(\lim_{x \to c} f(x) = L\) contains several parts:

lim — short for "limit"
\(x \to c\) — "as \(x\) approaches \(c\)" (written below the lim)
\(f(x)\) — the function we're analyzing
\(= L\) — the value the function approaches

Here are some examples to practice reading:

Notation	How to Read It
\(\lim_{x \to 3} f(x) = 7\)	"The limit of \(f(x)\) as \(x\) approaches 3 equals 7"
\(\lim_{t \to 0} g(t) = -2\)	"The limit of \(g(t)\) as \(t\) approaches 0 equals negative 2"
\(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)	"The limit as \(h\) approaches 0 of the difference quotient"

The last one is important—it's the definition of the derivative, which you'll meet soon!

The Target vs. The Value

A crucial distinction: in \(\lim_{x \to c} f(x) = L\), the value \(c\) is the target that \(x\) approaches, while \(L\) is the limit value that \(f(x)\) approaches.

These can be the same number, but they don't have to be. They're measuring different things:

\(c\) is about where we're going on the x-axis
\(L\) is about what's happening on the y-axis as we get there

One-Sided Limits

Sometimes a function behaves differently depending on which direction you approach from. This is where one-sided limits become essential.

Left-Hand Limits

A left-hand limit (or limit from the left) describes what \(f(x)\) approaches as \(x\) gets closer to \(c\) from values less than \(c\). We use a superscript minus sign:

\[\lim_{x \to c^-} f(x)\]

This reads: "The limit of \(f(x)\) as \(x\) approaches \(c\) from the left."

Delta Moment

"I'm rolling toward \(x = c\) from the left side of the number line. What y-value am I heading toward? That's my left-hand limit!"

Right-Hand Limits

A right-hand limit (or limit from the right) describes what \(f(x)\) approaches as \(x\) gets closer to \(c\) from values greater than \(c\). We use a superscript plus sign:

\[\lim_{x \to c^+} f(x)\]

This reads: "The limit of \(f(x)\) as \(x\) approaches \(c\) from the right."

Example: Piecewise Function

Consider this piecewise function:

\[f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ x^2 - 1 & \text{if } x \geq 2 \end{cases}\]

What are the one-sided limits at \(x = 2\)?

From the left: As \(x \to 2^-\), we use \(f(x) = x + 1\), so: \(\(\lim_{x \to 2^-} f(x) = 2 + 1 = 3\)\)

From the right: As \(x \to 2^+\), we use \(f(x) = x^2 - 1\), so: \(\(\lim_{x \to 2^+} f(x) = 4 - 1 = 3\)\)

Both one-sided limits equal 3! This leads us to two-sided limits.

Two-Sided Limits and Limit Existence

The Two-Sided Limit

When we write \(\lim_{x \to c} f(x)\) without any plus or minus sign, we mean the two-sided limit. This is the "regular" limit you'll use most often.

Here's the key rule:

The Two-Sided Limit Existence Condition

The two-sided limit \(\lim_{x \to c} f(x) = L\) exists if and only if both one-sided limits exist AND are equal:

\[\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L\]

In our piecewise example above, since both one-sided limits equal 3, the two-sided limit exists and equals 3:

\[\lim_{x \to 2} f(x) = 3\]

When Limits Fail to Exist

Limits can fail to exist for several reasons:

One-sided limits differ: If \(\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)\), the two-sided limit doesn't exist.
Function oscillates: Some functions wiggle infinitely as \(x \to c\) without settling on any value.
Function goes to infinity: We'll explore this case in Chapter 5.

Let's see an example where one-sided limits disagree.

Diagram: One-Sided Limits Comparison

One-Sided Limits Comparison MicroSim

Type: microsim

Bloom Level: Analyze (L4) Bloom Verb: Compare

Learning Objective: Students will compare left-hand and right-hand limits to determine whether a two-sided limit exists.

Visual elements:

Coordinate plane with grid
Piecewise function with a jump discontinuity
Two Delta robots: one approaching from left (teal), one from right (orange)
Dashed horizontal lines showing the y-values each Delta is approaching
Clear visual gap between the two limit values
Display panel showing: Left limit, Right limit, and "Two-sided limit exists?" (Yes/No)

Interactive controls:

Function selector dropdown with 4 preset piecewise functions:
Function A: Jump at x = 1 (limits differ)
Function B: Continuous at x = 2 (limits equal)
Function C: Removable discontinuity (limits equal)
Function D: Jump at x = 0 (limits differ)
Slider: Approach distance (0.5 to 0.01)
Button: "Animate both Deltas"
Button: "Reset"

Behavior:

Both Deltas move toward the target x-value simultaneously
Y-value readouts update in real-time
When limits differ, a red "X" appears and "Does not exist" displays
When limits match, a green checkmark appears with the limit value

Instructional Rationale: Comparing left and right approaches simultaneously makes the limit existence criterion concrete. Students see that disagreeing one-sided limits prevent the two-sided limit from existing.

Data Visibility Requirements:

Stage 1: Show both Deltas at starting positions with initial y-values
Stage 2: Show both approaching, y-values converging or diverging
Stage 3: Show final approach with numerical comparison
Final: Display conclusion about limit existence

Implementation: p5.js with canvas-based controls

The Limit Laws

You now understand what limits mean. But evaluating limits by making tables every time would be tedious. Fortunately, mathematicians have proven limit laws that let us compute limits algebraically.

These laws work whenever both \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) exist.

Constant Multiple Rule

\[\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)\]

You can pull constants out of limits. If \(f(x)\) approaches \(L\), then \(3f(x)\) approaches \(3L\).

Sum Rule for Limits

\[\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)\]

The limit of a sum equals the sum of the limits.

Difference Rule

\[\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)\]

The limit of a difference equals the difference of the limits.

Product Rule for Limits

\[\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)\]

The limit of a product equals the product of the limits.

Quotient Rule for Limits

\[\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}, \quad \text{provided } \lim_{x \to c} g(x) \neq 0\]

The limit of a quotient equals the quotient of the limits—as long as the denominator's limit isn't zero!

Power Rule for Limits

\[\lim_{x \to c} [f(x)]^n = \left[\lim_{x \to c} f(x)\right]^n\]

The limit of a power equals the power of the limit.

Summary Table of Limit Laws

Law	Formula
Constant Multiple	\(\lim [k \cdot f] = k \cdot \lim f\)
Sum	\(\lim [f + g] = \lim f + \lim g\)
Difference	\(\lim [f - g] = \lim f - \lim g\)
Product	\(\lim [f \cdot g] = \lim f \cdot \lim g\)
Quotient	\(\lim \frac{f}{g} = \frac{\lim f}{\lim g}\) (if \(\lim g \neq 0\))
Power	\(\lim [f]^n = [\lim f]^n\)

Direct Substitution

Here's wonderful news: for most functions you encounter, finding limits is surprisingly simple.

The Direct Substitution Property

If \(f(x)\) is a polynomial, rational function (where the denominator isn't zero), or any combination of continuous functions, then:

\[\lim_{x \to c} f(x) = f(c)\]

Just plug in the value! This is called direct substitution.

Examples of Direct Substitution

Example 1: Find \(\lim_{x \to 3} (2x^2 - 5x + 1)\)

This is a polynomial, so we can substitute directly: \(\(\lim_{x \to 3} (2x^2 - 5x + 1) = 2(3)^2 - 5(3) + 1 = 18 - 15 + 1 = 4\)\)

Example 2: Find \(\lim_{x \to 2} \frac{x + 1}{x - 5}\)

This is a rational function. At \(x = 2\), the denominator is \(2 - 5 = -3 \neq 0\), so we can substitute: \(\(\lim_{x \to 2} \frac{x + 1}{x - 5} = \frac{2 + 1}{2 - 5} = \frac{3}{-3} = -1\)\)

Example 3: Find \(\lim_{\theta \to \pi/4} \sin(\theta)\)

Sine is continuous everywhere, so: \(\(\lim_{\theta \to \pi/4} \sin(\theta) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)\)

When Direct Substitution Fails

Direct substitution doesn't work when it produces an undefined expression, like:

\(\frac{0}{0}\) — indeterminate form (we'll handle this in Chapter 3)
\(\frac{\text{nonzero}}{0}\) — possible infinite limit (Chapter 5)

When direct substitution fails, you'll need the techniques from the next chapter.

Diagram: Direct Substitution Decision Tree

Direct Substitution Decision Tree

Type: workflow

Bloom Level: Apply (L3) Bloom Verb: Use

Learning Objective: Students will use a systematic decision process to determine whether direct substitution can evaluate a limit.

Purpose: Guide students through the decision process for evaluating limits.

Visual style: Flowchart with decision diamonds and process rectangles

Steps:

Start: "Given \(\lim_{x \to c} f(x)\)" Hover text: "We want to find the limit of f(x) as x approaches c"
Process: "Try substituting x = c into f(x)" Hover text: "Calculate f(c) directly"
Decision: "Does f(c) produce a real number?" Hover text: "Check if the result is defined (not 0/0, not nonzero/0)"

4a. Process: "Limit = f(c)" (if YES) Hover text: "Direct substitution works! The limit equals f(c)."

4b. Decision: "Is the result 0/0?" (if NO) Hover text: "0/0 is an indeterminate form that requires algebraic techniques"

5a. Process: "Use algebraic techniques (Chapter 3)" Hover text: "Try factoring, rationalization, or other methods"

5b. Decision: "Is the result nonzero/0?" (if not 0/0) Hover text: "This suggests the limit may be infinite or not exist"

Process: "Investigate infinite limit (Chapter 5)" Hover text: "Check one-sided behavior to determine if limit is +∞, -∞, or DNE"

Color coding:

Green: Direct substitution success
Yellow: Decision points
Orange: Need algebraic techniques
Red: Need asymptote analysis

Implementation: p5.js with interactive flowchart, highlight current step on hover

Putting It All Together

Let's work through a comprehensive example using limit laws.

Problem: Find \(\lim_{x \to 2} \frac{3x^2 + 2x - 1}{x + 4}\)

Solution:

First, check if direct substitution works. At \(x = 2\):

Denominator: \(2 + 4 = 6 \neq 0\) ✓

Direct substitution is valid! Using the quotient rule:

\[\lim_{x \to 2} \frac{3x^2 + 2x - 1}{x + 4} = \frac{\lim_{x \to 2}(3x^2 + 2x - 1)}{\lim_{x \to 2}(x + 4)}\]

For the numerator, apply sum and constant multiple rules: \(\(\lim_{x \to 2}(3x^2 + 2x - 1) = 3(2)^2 + 2(2) - 1 = 12 + 4 - 1 = 15\)\)

For the denominator: \(\(\lim_{x \to 2}(x + 4) = 2 + 4 = 6\)\)

Therefore: \(\(\lim_{x \to 2} \frac{3x^2 + 2x - 1}{x + 4} = \frac{15}{6} = \frac{5}{2}\)\)

Delta's Pun Corner

"Why did the limit go to therapy? It had trouble approaching its problems. ...I'll see myself out."

Practice: Building Your Limit Intuition

Diagram: Limit Laws Practice

Limit Laws Practice MicroSim

Type: microsim

Bloom Level: Apply (L3) Bloom Verb: Calculate

Learning Objective: Students will calculate limits by selecting and applying appropriate limit laws.

Visual elements:

Problem display area showing a limit expression
Step-by-step workspace showing the solution process
Available limit laws displayed as clickable cards
Running calculation showing intermediate results
Score tracker for correct answers

Interactive controls:

Button: "New Problem" generates a random limit problem
Difficulty selector: "Basic," "Intermediate," "Challenge"
Clickable law cards that students select to apply
Input field for final numerical answer
Button: "Check Answer"
Button: "Show Solution"

Problem types by difficulty:

Basic: Single rule applications (e.g., \(\lim_{x \to 3} 5x\))
Intermediate: Two rules combined (e.g., \(\lim_{x \to 2} (x^2 + 3x)\))
Challenge: Three or more rules (e.g., \(\lim_{x \to 1} \frac{2x^2 - x}{x + 1}\))

Behavior:

Each problem shows the expression and target x-value
Students click law cards in the order they'd apply them
System highlights whether the sequence is valid
Final answer is checked numerically
"Show Solution" reveals step-by-step worked solution

Instructional Rationale: Active practice with immediate feedback reinforces procedural fluency while the law-selection interface makes the reasoning process explicit.

Implementation: p5.js with canvas-based controls and problem generation

Key Takeaways

Let's summarize what you've learned in this chapter:

A limit describes what value a function approaches as its input approaches some target—regardless of what happens at that exact point.
Limit notation \(\lim_{x \to c} f(x) = L\) means "\(f(x)\) approaches \(L\) as \(x\) approaches \(c\)."
One-sided limits look at approach from only one direction:
Left-hand limit: \(\lim_{x \to c^-} f(x)\)
Right-hand limit: \(\lim_{x \to c^+} f(x)\)
A two-sided limit exists only when both one-sided limits exist and are equal.
Limit laws let you break apart complex limits:
Sum, difference, product, quotient rules
Constant multiple and power rules
Direct substitution works for most functions: if \(f\) is continuous at \(c\), then \(\lim_{x \to c} f(x) = f(c)\).

Delta Moment

"Limits are about the journey, not the destination. What matters is where you're headed, even if you never quite arrive. That's actually pretty deep for math, right?"

Looking Ahead

You've built a solid foundation for understanding limits. But what happens when direct substitution gives you \(\frac{0}{0}\)? In Chapter 3, you'll learn algebraic techniques to handle these tricky cases, including the powerful Squeeze Theorem.

Check Your Understanding

What's the difference between \(f(2)\) and \(\lim_{x \to 2} f(x)\)?
If \(\lim_{x \to 5^-} f(x) = 3\) and \(\lim_{x \to 5^+} f(x) = 7\), what is \(\lim_{x \to 5} f(x)\)?
True or false: If direct substitution gives a number, that number is the limit.

Answers

\(f(2)\) is the actual value of the function at \(x = 2\), while \(\lim_{x \to 2} f(x)\) is the value the function approaches as \(x\) gets close to 2. These can be different (or \(f(2)\) might not even exist).
The limit does not exist because the one-sided limits are different.
True! When direct substitution produces a real number, that is the limit value.

See Annotated References