Always substitute first. For a continuous function the limit is the value, and no further work is needed.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The indeterminate form
0 ÷ 0 is not an answer. It reports that the top and bottom share a factor, which must be removed before the limit appears.
Factor and cancel
Factor both parts, cancel the shared factor, then substitute into what remains.
When a root is in the way
For expressions with a square root, multiply by the conjugate. The difference of squares clears the root and exposes the shared factor.
Stacked fractions
A compound fraction is cleared by multiplying through by the common denominator, which usually reveals the cancellation.
Not all zeros are equal
A non-zero number over zero is not indeterminate. It means the values run away, and the limit does not exist.
Substitution works for continuous functions
If the function is continuous at the point, the limit is the value. Most limits are settled this way in one step, so it is always the first thing to try.
What 0/0 actually means
An indeterminate form does not mean the limit fails to exist — it means substitution has not decided it. Different functions of that form have different limits, which is precisely why more work is needed.
Factor, conjugate, simplify
Factor and cancel the offending factor; multiply by a conjugate to clear a radical; combine a complex fraction. Each technique removes the source of the zero denominator, revealing the limit underneath.
The cancelled factor was the hole
Cancelling (x − 2) from a quotient produces a function agreeing with the original everywhere except at x = 2, where the original had a hole. Since limits ignore the point itself, the cancellation is legitimate.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 2)
Step 3: Watch an Example
One step at a time.
Watch Priya Rationalise a Limit
Priya needs the limit of (√(x + 4) − 2) ÷ x as x approaches 0.
- Step 1
Substituting 0 gives (2 − 2) ÷ 0, which is the indeterminate form 0 ÷ 0.
Step 4: Your Turn
Practice makes it stick.
The Cancel
Problem 1 of 2
Limit of (x² − 9) ÷ (x − 3) as x approaches 3?
The Substitution
Problem 2 of 2
Limit of (3x + 5) as x approaches 2?
Clear the Obstruction
1 of 8
Limit of (x² − 4) ÷ (x − 2) as x approaches 2?
2 of 8
Limit of (x² − 25) ÷ (x − 5) as x approaches 5?
3 of 8
Limit of 5x − 1 as x approaches 4?
4 of 8
Limit of (x² − x − 6) ÷ (x − 3) as x approaches 3?
5 of 8
Limit of (x³ − 8) ÷ (x − 2) as x approaches 2?
6 of 8
Limit of (√(x + 9) − 3) ÷ x as x approaches 0, as a decimal?
7 of 8
Sort each form by what it tells you.
Tap something to move it.
- Empty
- Empty
8 of 8
Limit of (x² − 16) ÷ (x + 4) as x approaches −4?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Limit of (x² − 1) ÷ (x − 1) as x approaches 1?
Question 2 of 2
Substituting gives 0 ÷ 0. What does that mean?
What You Learned
- Substitute first; if it gives a number, that is the limit.
- 0 ÷ 0 means factor, rationalise, or clear the fractions, then substitute.
- A non-zero number over zero means the limit does not exist.