A limit asks what value a function approaches as the input nears a number, whether or not the function is defined there.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A hole changes nothing
f(x) = (x² − 1) ÷ (x − 1) is undefined at x = 1, yet approaches 2 from both sides. The limit is 2.
Limit and value can differ
A function may equal 7 at x = 3 while approaching 5 there. Both statements can be true at once.
One-sided limits
The left-hand limit approaches from below, the right-hand from above. The two-sided limit exists only when they agree.
When a limit fails
A limit fails to exist at a jump, where the two sides disagree, or at a vertical asymptote, where the values run away.
Why it matters
Every idea in calculus is a limit underneath: the derivative is a limit of slopes and the integral is a limit of sums.
Where it is heading, not where it lands
The limit of f(x) as x approaches a describes what the outputs approach, regardless of what happens at a itself. The function may be undefined there and the limit can still exist.
Two sides must agree
The limit exists only if the left-hand and right-hand limits are equal. A jump discontinuity has both one-sided limits and no overall limit, which is exactly what makes it a jump.
A limit is not a function value
A function can have a hole at a point and a perfectly good limit there, or be defined at a point with a different limit. Conflating the two is the central misconception the concept is designed to separate.
Why the idea is needed
Instantaneous speed is a change over an interval that has shrunk to nothing — a division of zero by zero. Limits make that meaningful, and without them calculus cannot be stated at all.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Marcus Estimate a Limit
Marcus wants the limit of (x² − 1) ÷ (x − 1) as x approaches 1.
- Step 1
Substituting 1 gives 0 ÷ 0, which is undefined, so he cannot simply evaluate.
Step 4: Your Turn
Practice makes it stick.
The Approach
Problem 1 of 2
What is the limit of 2x + 1 as x approaches 3?
The Hole
Problem 2 of 2
What is the limit of (x² − 4) ÷ (x − 2) as x approaches 2?
What Is It Approaching?
1 of 8
Limit of 3x as x approaches 4?
2 of 8
Limit of x² as x approaches 5?
3 of 8
Limit of (x² − 9) ÷ (x − 3) as x approaches 3?
4 of 8
Limit of 7 as x approaches 2?
5 of 8
Left limit is 3 and right limit is 5. Does the two-sided limit exist? 1 for yes, 0 for no.
6 of 8
Limit of (x² − 25) ÷ (x − 5) as x approaches 5?
7 of 8
Which situations mean a limit does not exist?
8 of 8
Limit of x + 8 as x approaches −3?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Limit of 4x − 2 as x approaches 3?
Question 2 of 2
A function is undefined at x = 2. Can its limit at 2 exist?
What You Learned
- A limit describes where a function is heading, not where it lands.
- A limit can exist where the function is undefined.
- The two-sided limit exists only when both one-sided limits agree.