Introduction to Limits

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

Advanced25 minLesson
Printable worksheet

Definition

A limit describes the value that a function approaches as its input approaches a particular value.
We write:
lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
This reads: "The limit of f(x)f(x) as xx approaches aa equals LL."
Key idea: We're asking "What value does f(x)f(x) get close to as xx gets close to aa?" Even if f(a)f(a) itself is undefined!
Example: Consider f(x)=x2−1x−1f(x) = \frac{x^2 - 1}{x - 1}
  • At x=1x = 1: undefined (division by zero)
  • As x→1x \to 1: the function approaches 22
So we write: lim⁡x→1x2−1x−1=2\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2

Try it now

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

Worked Examples

Find lim⁡x→3(2x+1)\lim_{x \to 3} (2x + 1)
1

Check if direct substitution works

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

Substitute the value

f(3)=2(3)+1=6+1f(3) = 2(3) + 1 = 6 + 1 → =7= 7
3

Write the answer in limit notation

lim⁡x→3(2x+1)=7\lim_{x \to 3} (2x + 1) = 7 → The limit is 77

Common Mistakes

Confusing f(a)f(a) with lim⁡x→af(x)\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)f(x) get close to?' not 'What is f(a)f(a)?'

Saying a limit 'equals infinity' means it exists

Why it's wrong: When we write lim⁡=∞\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= L (exists, equals LL), =∞= \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→af(x)=L\lim_{x \to a} f(x) = L requires lim⁡x→a−f(x)=lim⁡x→a+f(x)=L\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

Canceling 00\frac{0}{0} incorrectly

Why it's wrong: 00\frac{0}{0} is indeterminate, not equal to 1 or 0. It signals that algebraic manipulation is needed.

Correct: When you get 00\frac{0}{0}, factor, rationalize, or use other techniques to simplify first.

Interactive Visual

Function graph

-3-2-101234-2024xy

Tap or hover over the graph to read off values.

f(x) = (x² − 1)/(x − 1)
  • At x = 1: from the left and from the right f(x) → 2, so the limit is 2. f(1) is undefined (a hole).

Table of values

x0.90.9911.011.1
y1.91.99undef.2.012.1

Function graph

-4-2024-505xyx = 0y = 0

Tap or hover over the graph to read off values.

y = 1/xAsymptote

Table of values

x-0.1-0.010.010.1
y-10-10010010

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = (x^2 - 1)/(x - 1)

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
What does lim⁡x→5f(x)=3\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)=t2s(t) = t^2 meters at time tt seconds, your instantaneous speed at t=3t = 3 is lim⁡h→0s(3+h)−s(3)h=lim⁡h→0(3+h)2−9h=6\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)=4t2s(t) = 4t^2 meters. You want to find the instantaneous speed at t=2t = 2 seconds.

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

Step 1: Write the mathematical expression

Set up and evaluate: lim⁡h→0s(2+h)−s(2)h\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)=K1+Ae−rtP(t) = \frac{K}{1 + Ae^{-rt}} shows that lim⁡t→∞P(t)=K\lim_{t \to \infty} P(t) = K, the carrying capacity.

2Try It Yourself

A population follows P(t)=10001+9e−0.5tP(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→∞P(t)\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→∞(1+1n)ne = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n represents continuous compounding.

3Try It Yourself

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

What value does this expression approach?

Step 1: Choose the right answer

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

Key Takeaways

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

Frequently Asked Questions

The function value f(a)f(a) is the output when you plug in aa. The limit lim⁡x→af(x)\lim_{x \to a} f(x) describes what the outputs approach as inputs get close to aa. These can be different! For example, a function might have a hole at x=2x = 2 (so f(2)f(2) is undefined) but still have a limit there.
The function value f(a)f(a) is the output when you plug in aa. The limit lim⁡x→af(x)\lim_{x \to a} f(x) describes what the outputs approach as inputs get close to aa. These can be different! For example, a function might have a hole at x=2x = 2 (so f(2)f(2) is undefined) but still have a limit there.
A limit DNE when there's no single value that f(x)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).
Because 00\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 xx approaches from only one direction (x→a+x \to a^+ or x→a−x \to a^-)
Indeterminate form
An expression like 00\frac{0}{0} or ∞∞\frac{\infty}{\infty} that requires further analysis
Continuous
A function where lim⁡x→af(x)=f(a)\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
Today's challengeOne problem, three tries

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