Introduction to Limits
A limit describes the value a function approaches as the input gets closer to a given point, forming the conceptual foundation for calculus and continuous mathematics.
What You Need to Know
Key Concept Diagram
The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to
Limits can be evaluated by direct substitution when the function is defined at that point
When direct substitution gives 0/0, factorise and cancel to resolve the indeterminate form
One-sided limits examine the value approached from the left or right separately
A function is continuous at a point if the limit equals the function value there
Key Vocabulary
Limit
The value a function approaches as the input approaches a specified value
Indeterminate form
An expression such as 0/0 that requires further manipulation to evaluate
Continuity
A function is continuous at a point if its limit there equals its function value
One-sided limit
The value approached by a function from only one direction (left or right)
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Evaluate the limit of (x^2 - 4)/(x - 2) as x approaches 2.
Question 2
Which condition must hold for f(x) to be continuous at x = a?
Question 3
What is the limit of (3x^2 + 2x)/(x) as x approaches 0?
Key Concepts Summary
- ●The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to
- ●Limits can be evaluated by direct substitution when the function is defined at that point
- ●When direct substitution gives 0/0, factorise and cancel to resolve the indeterminate form
- ●One-sided limits examine the value approached from the left or right separately
- ●A function is continuous at a point if the limit equals the function value there