What Is a Limit?

A limit describes the value that a function approaches as the input approaches a particular value. It does not matter what happens at that point — only what happens near it.

lim_x→a f(x) = L

Read as: "the limit of f(x) as x approaches a equals L"

This means that as x gets closer and closer to a (from both sides), the values of f(x) get closer and closer to L.

Example: f(x) = x² − 1x − 1 as x → 1

x	0.9	0.99	0.999	1	1.001	1.01	1.1
f(x)	1.9	1.99	1.999	undef	2.001	2.01	2.1

Even though f(1) is undefined, the values approach 2 from both sides. So lim_x→1 f(x) = 2.

Evaluating Limits Algebraically

There are several strategies for evaluating limits:

1

Direct substitution: Simply substitute x = a. If this gives a real number, that is the limit. Works when f is continuous at a.

2

Factorise and cancel: If substitution gives 0/0, factorise the numerator and denominator, cancel the common factor, then substitute.

3

Rationalise: If the expression involves surds, multiply by the conjugate to simplify.

The 0/0 Indeterminate Form

If direct substitution gives 0/0, it does NOT mean the limit is 0 or undefined. It means we need to simplify further before evaluating. This is called an indeterminate form.

One-Sided Limits

Sometimes we need to consider what happens as x approaches a value from one direction only:

lim_x→a⁻ f(x)

Left-hand limit (x approaches from below)

lim_x→a⁺ f(x)

Right-hand limit (x approaches from above)

The two-sided limit lim_x→a f(x) exists only if both one-sided limits exist and are equal: lim_x→a⁻ f(x) = lim_x→a⁺ f(x).

Key Vocabulary

Limit

The value a function approaches as the input gets closer to a specified value.

Continuous

A function is continuous at x = a if lim_x→a f(x) = f(a). No breaks or holes.

Indeterminate Form

An expression like 0/0 that requires further simplification to determine the limit.

One-Sided Limit

The limit as x approaches a from one direction only (left or right).

Worked Examples

1

Evaluate lim_x→3 (2x + 1).

Step 1: Try direct substitution: 2(3) + 1 = 7.

Answer: lim_x→3 (2x + 1) = 7.

2

Evaluate lim_x→2 x² − 4/x − 2.

Step 1: Direct substitution gives (4 − 4)/(2 − 2) = 0/0 — indeterminate.

Step 2: Factorise the numerator: x² − 4 = (x − 2)(x + 2).

Step 3: Cancel (x − 2): (x − 2)(x + 2)/x − 2 = x + 2.

Step 4: Substitute: 2 + 2 = 4.

3

Evaluate lim_x→4 √x − 2/x − 4.

Step 1: Direct substitution gives 0/0 — indeterminate.

Step 2: Multiply by the conjugate: √x − 2/x − 4 × √x + 2/√x + 2 = x − 4/(x − 4)(√x + 2)

Step 3: Cancel (x − 4): = 1/√x + 2.

Step 4: Substitute x = 4: 1/√4 + 2 = 1/4 = 0.25.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Evaluate lim_x→5 (3x − 2).

15 17 13 7

Question 2

Evaluate lim_x→1 x² − 1/x − 1.

0 1 Undefined 2

Question 3

What is the indeterminate form?

0/0 0/5 5/0 1/0

Question 4

For a two-sided limit to exist, what must be true?

The function must be defined at that point Both one-sided limits must exist and be equal The function must be a polynomial The limit must equal the function value

Question 5

Evaluate lim_x→3 x² − 9/x − 3.

0 3 6 9

Introduction to Limits

What Is a Limit?

Example: f(x) = x² − 1x − 1 as x → 1

Evaluating Limits Algebraically

The 0/0 Indeterminate Form

One-Sided Limits

Key Vocabulary

Limit

Continuous

Indeterminate Form

One-Sided Limit

Worked Examples

Evaluate lim_x→3 (2x + 1).

Evaluate lim_x→2 x² − 4/x − 2.

Evaluate lim_x→4 √x − 2/x − 4.

Knowledge Check

Key Concepts Summary

Introduction to Limits

What Is a Limit?

Example: f(x) = x² − 1x − 1 as x → 1

Evaluating Limits Algebraically

The 0/0 Indeterminate Form

One-Sided Limits

Key Vocabulary

Limit

Continuous

Indeterminate Form

One-Sided Limit

Worked Examples

Evaluate limx→3 (2x + 1).

Evaluate limx→2 x² − 4/x − 2.

Evaluate limx→4 √x − 2/x − 4.

Knowledge Check

Key Concepts Summary

Evaluate lim_x→3 (2x + 1).

Evaluate lim_x→2 x² − 4/x − 2.

Evaluate lim_x→4 √x − 2/x − 4.