Teacher Guide: Introduction to Limits

Learn the foundational concept of limits and how they describe the behavior of functions as inputs approach specific values.

Present LessonView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

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

For Teachers

Learning Objectives

Explain the intuitive meaning of a limit using precise language
Evaluate limits using direct substitution when applicable
Identify and resolve indeterminate forms using algebraic techniques
Distinguish between one-sided and two-sided limits
Calculate limits at infinity and interpret horizontal asymptotes
Connect limits to real-world contexts like instantaneous rates

Prerequisites

• Understanding of functions and function notation
• Ability to factor polynomials
• Familiarity with graphing functions
• Knowledge of rational expressions
• Basic understanding of infinity as a concept

Discussion Starters

1. Can a function have a limit at a point where it's not defined? Give an example.
2. Why do you think limits became so important for developing calculus?
3. What real-world situations involve values 'approaching' but never quite reaching something?
4. How would you explain to a friend why $\lim_{x \to 0} \frac{\sin x}{x} = 1$ ?

Common Misconceptions

The limit equals the function value

Remediation: Show examples like $f(x) = \frac{x^2-1}{x-1}$ where the limit at $x=1$ is $2$ , but $f(1)$ is undefined. Emphasize 'approaching' vs 'reaching'.

If I get $\frac{0}{0}$ , the limit is $0$ or $1$

Remediation: Show multiple examples where $\frac{0}{0}$ leads to different answers: $\lim_{x \to 0} \frac{x}{x} = 1$ , $\lim_{x \to 0} \frac{2x}{x} = 2$ , $\lim_{x \to 0} \frac{x^2}{x} = 0$ .

Infinity is a number you can calculate with

Remediation: Clarify that $\infty$ describes unbounded behavior, not a specific value. $\infty - \infty$ is not $0$ , and $\frac{\infty}{\infty}$ is not $1$ .

Differentiation Ideas

For Struggling Students:

• Start with limits of linear functions where direct substitution always works
• Use tables of values to show approaching behavior numerically
• Provide graph paper for students to visualize limits at holes

For On-Level Students:

• Practice factoring techniques for $\frac{0}{0}$ indeterminate forms
• Explore one-sided limits graphically and algebraically
• Connect limits to ideas of continuity

For Advanced Students:

• Introduce the formal epsilon-delta definition of limits
• Explore L'Hôpital's Rule for indeterminate forms
• Investigate limits involving trigonometric functions like $\frac{\sin x}{x}$

Standards Alignment

HSF-IF.C.7 (CCSS.MATH.CONTENT.HSF.IF.C.7)
Graph functions expressed symbolically and show key features of the graph
HSF-BF.B.3 (CCSS.MATH.CONTENT.HSF.BF.B.3)
Identify the effect on the graph of replacing f(x) with transformations

Lesson Resources

visualInteractive Limit Explorer
Graph functions and see values approach limits visually
activityHole vs. Value Game
Match functions with their limits and actual values
worksheetIndeterminate Forms Practice
Factor and simplify to find limits with holes

Lesson Content

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

Definition

A limit describes the value that a function approaches as its input approaches a particular value.

We write:

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

This reads: "The limit of

f(x)

x

approaches

a

equals

L

Key idea: We're asking "What value does

f(x)

get close to as

x

gets close to

a

?" — even if

f(a)

itself is undefined!

Example: Consider

f(x) = \frac{x^2 - 1}{x - 1}

At $x = 1$ : undefined (division by zero)
As $x \to 1$ : the function approaches $2$

So we write:

\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2

Worked Examples

Find $\lim_{x \to 3} (2x + 1)$

Check if direct substitution works

The function $f(x) = 2x + 1$ is defined at $x = 3$ → Direct substitution is valid

Substitute the value

$f(3) = 2(3) + 1 = 6 + 1$ → $= 7$

Write the answer in limit notation

$\lim_{x \to 3} (2x + 1) = 7$ → The limit is $7$

Answer: $\lim_{x \to 3} (2x + 1) = 7$

Common Mistakes

Confusing $f(a)$ with $\lim_{x \to a} f(x)$

Why it's wrong: The value of a function AT a point can differ from (or not exist while) the limit EXISTS. Limits describe approaching behavior, not the actual value.

Correct: Always think: 'What does $f(x)$ get close to?' not 'What is $f(a)$ ?'

Saying a limit 'equals infinity' means it exists

Why it's wrong: When we write $\lim = \infty$ , we're describing unbounded growth. Technically, the limit 'does not exist' as a finite number, but we use infinity notation to describe the behavior.

Correct: Distinguish between: DNE (doesn't exist), $= L$ (exists, equals $L$ ), $= \infty$ (unbounded)

Forgetting to check both sides for existence

Why it's wrong: A two-sided limit exists only if both one-sided limits exist AND are equal.

Correct: $\lim_{x \to a} f(x) = L$ requires $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

Canceling $\frac{0}{0}$ incorrectly

Why it's wrong: $\frac{0}{0}$ is indeterminate, not equal to 1 or 0. It signals that algebraic manipulation is needed.

Correct: When you get $\frac{0}{0}$ , factor, rationalize, or use other techniques to simplify first.

Why It Matters

Limits are the foundation of calculus and unlock powerful mathematical tools:

Derivatives: The instantaneous rate of change is defined using limits
Integrals: Areas under curves are computed using limits
Continuity: Whether a function has gaps depends on limits
Infinity: Limits let us rigorously discuss infinite behavior

Real applications:

Physics: instantaneous velocity and acceleration
Engineering: stress analysis at critical points
Economics: marginal cost and revenue
Computer Science: algorithm complexity analysis

Without limits, we couldn't describe motion at a single instant or calculate exact areas of curved shapes!

Real World Applications

Instantaneous Speed

When you check your speedometer, you see your instantaneous speed — not your average speed. This is calculated using limits.

Example:

If your position is $s(t) = t^2$ meters at time $t$ seconds, your instantaneous speed at $t = 3$ is $\lim_{h \to 0} \frac{s(3+h) - s(3)}{h} = \lim_{h \to 0} \frac{(3+h)^2 - 9}{h} = 6$ m/s

1Try It Yourself

A car's position is given by $s(t) = 4t^2$ meters. You want to find the instantaneous speed at $t = 2$ seconds.

What is the car's instantaneous speed at $t = 2$ ?

Step 1: Write the mathematical expression

Set up: $\lim_{h \to 0} \frac{s(2+h) - s(2)}{h}$

Population Growth Models

Biologists use limits to model carrying capacity — the maximum population an environment can sustain.

Example:

The logistic model $P(t) = \frac{K}{1 + Ae^{-rt}}$ shows that $\lim_{t \to \infty} P(t) = K$ , the carrying capacity.

2Try It Yourself

A population follows $P(t) = \frac{1000}{1 + 9e^{-0.5t}}$

What is the carrying capacity (long-term population limit)?

Step 1: Write the mathematical expression

Find $\lim_{t \to \infty} P(t)$

Compound Interest and $e$

The number $e \approx 2.718$ comes from a limit involving compound interest calculated infinitely often.

Example:

$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$ represents continuous compounding.

3Try It Yourself

Calculate $\left(1 + \frac{1}{n}\right)^n$ for increasing values of $n$ .

What value does this expression approach?

Step 1: Write the mathematical expression

Evaluate for $n = 1, 10, 100, 1000$

Key Takeaways

1 $\lim_{x \to a} f(x) = L$ means $f(x)$ approaches $L$ as $x$ approaches $a$
2The limit may exist even if $f(a)$ is undefined
3One-sided limits: $\lim_{x \to a^+}$ (from right) and $\lim_{x \to a^-}$ (from left)
4Two-sided limit exists only if both one-sided limits exist and are equal
5 $\frac{0}{0}$ is indeterminate — use factoring, rationalization, or other techniques
6Limits at infinity describe end behavior: $\lim_{x \to \infty} f(x)$

Frequently Asked Questions

What's the difference between a limit and the function value?

The function value

f(a)

is the output when you plug in

a

. The limit

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

describes what the outputs approach as inputs get close to

a

. These can be different! For example, a function might have a hole at

x = 2

(so

f(2)

is undefined) but still have a limit there.

What does 'does not exist' mean for a limit?

A limit DNE when there's no single value that

f(x)

approaches. This happens when: (1) left and right limits are different, (2) the function oscillates wildly, or (3) we say '

= \infty

' to indicate unbounded growth (technically DNE as a finite value).

Why is $\frac{0}{0}$ called 'indeterminate'?

Because

\frac{0}{0}

doesn't determine a specific value — the actual limit could be any number or even infinity. It just signals that the numerator and denominator both vanish, and you need to dig deeper (factor, simplify, etc.) to find the true limit.

Glossary

Limit: The value a function approaches as its input approaches a specified value
One-sided limit: A limit where $x$ approaches from only one direction ( $x \to a^+$ or $x \to a^-$ )
Indeterminate form: An expression like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ that requires further analysis
Continuous: A function where $\lim_{x \to a} f(x) = f(a)$ for all points in its domain
Asymptote: A line that a graph approaches but never reaches (horizontal, vertical, or oblique)
DNE: Does Not Exist — used when a limit has no defined value

Teacher Guide: Introduction to Limits

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

Instantaneous Speed

Population Growth Models

Compound Interest and $e$

Key Takeaways

Frequently Asked Questions

Glossary

More in This Topic

Limit Notation