Introduction to Limits

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

Advanced25 minLesson

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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

Try it now

What does

\lim_{x \to 5} f(x) = 3

mean?

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.

Interactive Visual

Linear Function Explorer

y = x

Slope (m)1

Y-Intercept (b)0

run

rise

x	y
-2	-2
-1	-1
0	0
1	1
2	2

Sequence Explorer

Sequence Type

First Term (a₁ = 2)

Common Difference (d = 3)

11,

14,

17,

20,

23,

26,

, ...

Formula

a_n = 2 + 3(n - 1) = 2 + 3n - 3

Sum Formula: S_10 = 10/2 × (2×2 + 9×3) = 155

a_1

a_5

a_10

a_20

Interactive Sandbox

Interactive Grapher

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

What does $\lim_{x \to 5} f(x) = 3$ mean?

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

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.

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

Introduction to Limits

Definition

Worked Examples

Common Mistakes

Interactive Visual

Linear Function Explorer

Sequence Explorer

Formula

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Instantaneous Speed

Population Growth Models

Compound Interest and $e$

Key Takeaways

Frequently Asked Questions

Glossary

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