A limit asks what the outputs approach as the inputs approach a point, never what happens at the point itself. A function can be undefined at 2, or defined as something absurd, and still have a limit of 4 there. Both sides must agree.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Watch it approach
x = 1.9, 1.99, 1.999 give 3.9, 3.99, 3.999. x = 2.1, 2.01, 2.001 give 4.1, 4.01, 4.001. From both sides the values close in on 4.
The value there is irrelevant
Define f(2) = 100 and the limit at 2 is still 4. The limit is about the neighbors; the value is about the point. The exam tests exactly this distinction.
One-sided limits
x → 2⁻ approaches from the left, x → 2⁺ from the right. For f(x) = x + 1 when x < 2 and f(x) = 2x when x ≥ 2: left limit 3, right limit 4. They differ, so the limit does not exist.
Other failures
1/(x − 2) as x → 2: the values run away, +∞ from the right, −∞ from the left. No single number: the limit does not exist. sin(1/x) as x → 0 oscillates forever: no limit either.
Writing it
lim as x → 2 of (x² − 4)/(x − 2) = 4. The arrow is the whole idea: approaching, never arriving. Write DNE, does not exist, when there is no limit.
Why this is the foundation
The derivative is a limit of slopes, the integral a limit of sums. Both are 0/0 or ∞ × 0 at the point, and only the limit gets past that. Everything after this rests on it.
Step 2: Try It Yourself
Tap and try it out.
f(x) = x + 1 for x < 2, f(2) = 7, and f(x) = 2x for x > 2. Find the limit at 2, and at 1
- 1x → 2⁻: x + 1 → 3the left piece
Step 0 of 4
- Point(2, 0.50)
Step 3: In Real Life
A speedometer
A speedometer reports a limit: distance over an ever-shorter interval. No interval is zero long, yet the needle points at a number. That is where the ratio is heading.
Step 4: 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 5: 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?
4 of 8
Left limit is 2, right limit is 5. Does the limit exist?
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 6: 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.
- The derivative and the integral are both limits.