A limit asks what the outputs approach as the inputs approach a point — never what happens at the point itself.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The value there is irrelevant
A function can be undefined at 2, or defined as something absurd, and still have a limit of 5 as x approaches 2.
Both sides must agree
If the left-hand and right-hand limits differ, the limit does not exist.
Why this is the foundation
The derivative and the integral are both limits. Everything after this rests on it.
Heading, not landing
The limit describes what the outputs approach as the input nears a value, regardless of the function value there. A function can be undefined at the point and still have a perfectly good limit.
Both sides must agree
A two-sided limit exists only when the left and right limits are equal. A jump has two good one-sided limits and no overall limit, which is exactly what makes it a jump.
Notation the exam expects
Write lim as x→a of f(x) = L, with the arrow and the value stated. Free-response answers lose credit for describing a limit in words when the notation was asked for, and vice versa.
Infinity is not a value
Saying a limit "equals infinity" is shorthand for the function growing without bound. The limit does not exist in the strict sense, and answers should say the function increases without bound rather than that it reaches infinity.
Step 2: Try It Yourself
Move the point along the curve and watch the outputs settle.
- Point(2, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Nadia Evaluate a Removable Case
Nadia finds the limit of (x² − 4)/(x − 2) as x approaches 2.
- Step 1
Substituting 2 gives 0/0, which says nothing yet.
Step 4: Your Turn
Practice makes it stick.
The Removable Discontinuity
Problem 1 of 2
What is the limit of (x² − 9)/(x − 3) as x approaches 3?
Value Against Limit
Problem 2 of 2
f(2) = 10 but the outputs near 2 approach 3. What is the limit as x approaches 2?
Approaching
1 of 8
Limit of 3x + 1 as x approaches 2?
2 of 8
Limit of (x² − 1)/(x − 1) as x approaches 1?
3 of 8
Left limit is 4, right limit is 4. Does the limit exist? 1 for yes, 0 for no.
4 of 8
Left limit is 2, right limit is 5. Does the limit exist? 1 for yes, 0 for no.
5 of 8
Select every statement that is true about limits.
6 of 8
Limit of 5 as x approaches 100?
7 of 8
Limit of (x² − 25)/(x − 5) as x approaches 5?
8 of 8
Order these steps for a limit that gives 0/0 on substitution.
- 1Factor the numerator and denominator
- 2Cancel the shared factor
- 3Substitute again
- 4Substitute and get 0/0
Step 5: Quick Check
Show what you know.
Question 1 of 1
Limit of (x² − 16)/(x − 4) as x approaches 4?
What You Learned
- A limit describes what the outputs approach, not what happens at the point.
- A limit exists only when both one-sided limits agree.
- 0/0 means more work is needed, not that the answer is zero.