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:
This reads: "The limit of as approaches equals ."
Key idea: We're asking "What value does get close to as gets close to ?" — even if itself is undefined!
Example: Consider
  • At : undefined (division by zero)
  • As : the function approaches
So we write:

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What does mean?

Worked Examples

Find

1

Check if direct substitution works

The function is defined at → Direct substitution is valid

2

Substitute the value

→

3

Write the answer in limit notation

→ The limit is

Common Mistakes

Confusing with

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 get close to?' not 'What is ?'

Saying a limit 'equals infinity' means it exists

Why it's wrong: When we write , 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), (exists, equals ), (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: requires

Canceling incorrectly

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

Correct: When you get , factor, rationalize, or use other techniques to simplify first.

Interactive Visual

Linear Function Explorer

y = x
Slope (m)1
Y-Intercept (b)0
b
run
rise
xy
-2-2
-1-1
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Sequence Explorer

2,
5,
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Formula

a_n = 2 + 3(n - 1) = 2 + 3n - 3
Sum Formula: S_10 = 10/2 × (2×2 + 9×3) = 155
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a_1

2

a_5

14

a_10

29

a_20

59

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y = 2x + 1

m=2, b=1

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Practice Problems

18 problems
Problem 1 of 18
Easy

What does 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 meters at time seconds, your instantaneous speed at is m/s

1Try It Yourself

A car's position is given by meters. You want to find the instantaneous speed at seconds.

What is the car's instantaneous speed at ?

Step 1: Write the mathematical expression

Set up:

Population Growth Models

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

Example:

The logistic model shows that , the carrying capacity.

2Try It Yourself

A population follows

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

Step 1: Write the mathematical expression

Find

Compound Interest and $e$

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

Example:

represents continuous compounding.

3Try It Yourself

Calculate for increasing values of .

What value does this expression approach?

Step 1: Write the mathematical expression

Evaluate for

Key Takeaways

  • 1 means approaches as approaches
  • 2The limit may exist even if is undefined
  • 3One-sided limits: (from right) and (from left)
  • 4Two-sided limit exists only if both one-sided limits exist and are equal
  • 5 is indeterminate — use factoring, rationalization, or other techniques
  • 6Limits at infinity describe end behavior:

Frequently Asked Questions

The function value is the output when you plug in . The limit describes what the outputs approach as inputs get close to . These can be different! For example, a function might have a hole at (so is undefined) but still have a limit there.
The function value is the output when you plug in . The limit describes what the outputs approach as inputs get close to . These can be different! For example, a function might have a hole at (so is undefined) but still have a limit there.
A limit DNE when there's no single value that approaches. This happens when: (1) left and right limits are different, (2) the function oscillates wildly, or (3) we say '' to indicate unbounded growth (technically DNE as a finite value).
Because 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 approaches from only one direction ( or )
Indeterminate form
An expression like or that requires further analysis
Continuous
A function where 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

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