For a continuous function, the limit is the value. Substituting is the whole method.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The 0 ÷ 0 case
0 ÷ 0 is called indeterminate: it says nothing about the answer, only that more work is needed.
Factor and cancel
The offending factor usually appears top and bottom. Cancelling it removes the hole, and substitution then works.
A number over zero
Something like 5 ÷ 0 is not indeterminate. It signals a vertical asymptote, where the limit does not exist.
Limits at infinity
As x grows without bound, a rational function is governed by its highest-degree terms alone.
Three outcomes
Equal degrees give the ratio of leading coefficients. A smaller top gives 0. A larger top gives no finite limit.
Try substitution first
For a continuous function, the limit is just the value: substitute and you are done. Most limits are this easy, and it is worth trying before reaching for any technique.
When substitution gives 0/0
A form like 0/0 is indeterminate — it does not mean the limit fails to exist, only that substitution has not settled it. Factoring and cancelling usually resolves it, and the cancelled factor is where the hole was.
The standard techniques
Factor and cancel; multiply by a conjugate to clear a radical; simplify a complex fraction; divide through by the highest power for limits at infinity. Four techniques cover almost every limit in a precalculus course.
Limits at infinity and infinite limits
A limit at infinity describes end behaviour and may be a finite number. An infinite limit means the function grows without bound, and saying the limit "is infinity" is shorthand for that — not a value being reached.
Step 2: Try It Yourself
Tap and try it out.
- Point(4, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Rosa Evaluate a Limit at Infinity
Rosa needs the limit of (3x² + 2x) ÷ (5x² − 1) as x grows without bound.
- Step 1
Substituting infinity gives an unusable form, so she compares degrees instead.
Step 4: Your Turn
Practice makes it stick.
The Cancel
Problem 1 of 2
Limit of (x² − 16) ÷ (x − 4) as x approaches 4?
The Far End
Problem 2 of 2
Limit of (4x² + 1) ÷ (2x² + 3) as x grows without bound?
Evaluate It
1 of 8
Limit of (x² − 1) ÷ (x − 1) as x approaches 1?
2 of 8
Limit of (x² − 36) ÷ (x − 6) as x approaches 6?
3 of 8
Limit of (6x² + x) ÷ (3x² − 2) as x grows without bound?
4 of 8
Limit of (x + 1) ÷ (x² + 3) as x grows without bound?
5 of 8
Limit of (10x² + 5) ÷ (4x² + 1) as x grows without bound?
6 of 8
Is 0 ÷ 0 an answer or a signal to do more work? Enter 1 for answer, 2 for signal.
7 of 8
Sort each degree comparison by the limit at infinity it produces.
Tap something to move it.
- Empty
- Empty
8 of 8
Limit of (x² − 49) ÷ (x − 7) as x approaches 7?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Limit of (x² − 4) ÷ (x − 2) as x approaches 2?
Question 2 of 2
What does the form 0 ÷ 0 tell you?
What You Learned
- For continuous functions, substitute and you are done.
- 0 ÷ 0 is indeterminate: factor, cancel, then substitute.
- At infinity, only the highest-degree terms of a rational function matter.