A limit asks where a function is heading as x approaches a number, not what it does there.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The two can differ
(x² − 1)/(x − 1) is undefined at x = 1, yet as x approaches 1 the value approaches 2.
Why this matters so much
The derivative is exactly this: a quantity that is undefined at the point and has a limit there.
One-sided limits
Approaching from the left and the right can give different answers. If they differ, the limit does not exist.
Heading, not arriving
The limit of f(x) as x approaches a describes the value the outputs get close to, without reference to f(a) itself. The function may be undefined at a and the limit can still exist perfectly well.
Both sides must agree
The limit exists only when the left-hand and right-hand limits are equal. A jump discontinuity has two perfectly good one-sided limits and no two-sided limit, which is exactly what the jump consists of.
The formal definition
For every tolerance ε on the output there is a tolerance δ on the input that keeps you within it. The definition is a promise about how close you can be forced to stay, and it is what made calculus rigorous after two centuries of use.
Why the concept was needed
Instantaneous rate of change is a ratio whose numerator and denominator both go to zero. Limits are the machinery that extracts a definite value from that, and without them the derivative cannot even be defined.
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 Priya Evaluate a Limit with a Hole
Priya finds the limit of (x² − 1)/(x − 1) as x approaches 1.
- Step 1
Substituting 1 gives 0/0, which says nothing.
Step 4: Your Turn
Practice makes it stick.
The Limit
Problem 1 of 2
The limit of (x² − 4)/(x − 2) as x approaches 2. What is it?
The Direct One
Problem 2 of 2
The limit of 3x + 1 as x approaches 5. What is it?
Where Is It Heading?
1 of 4
Limit of 2x − 3 as x approaches 4?
2 of 4
Limit of (x² − 9)/(x − 3) as x approaches 3?
3 of 4
Limit of (x² − 25)/(x − 5) as x approaches 5?
4 of 4
Must a function be defined at a point to have a limit there?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Limit of (x² − 36)/(x − 6) as x approaches 6?
Question 2 of 2
What does a limit describe?
What You Learned
- A limit is where a function heads, not what it equals.
- A function need not be defined at a point to have a limit there.
- If the two one-sided limits differ, the limit does not exist.