The big idea of calculus
Calculus studies how things change. Its foundation is the limit: what value a function approaches as x gets closer and closer to some number.
Notation: lim (x→a) f(x) = L
Read: "the limit of f(x) as x approaches a is L."
A first example
f(x) = 2x + 1. What happens as x approaches 3?
| x | 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 |
|---|---|---|---|---|---|---|
| f(x) | 6.8 | 6.98 | 6.998 | 7.002 | 7.02 | 7.2 |
From both sides, f(x) gets closer to 7. So lim (x→3) (2x + 1) = 7.
Limits don’t need the function at the point
f(x) = (x² − 4)/(x − 2) is undefined at x = 2 (division by 0). But:
| x | 1.9 | 1.99 | 2.01 | 2.1 |
|---|---|---|---|---|
| f(x) | 3.9 | 3.99 | 4.01 | 4.1 |
The limit as x → 2 is 4, even though f(2) doesn’t exist. (The graph is the line y = x + 2 with a hole at (2, 4).)
A limit describes where the function is heading, not where it is.
One-sided limits
A one-sided limit approaches from one side only.
- From the left: lim (x→a⁻) f(x)
- From the right: lim (x→a⁺) f(x)
The two-sided limit exists only if both one-sided limits exist and are equal.
Example: f(x) = 1 for x < 0 and 3 for x ≥ 0
- Left limit at 0: 1
- Right limit at 0: 3
- They differ → the limit does not exist
Infinite limits
f(x) = 1/x² as x → 0: values grow without bound.
lim (x→0) 1/x² = ∞ (the limit doesn’t exist as a number; the graph has a vertical asymptote).
Infinity isn’t a number; it describes unbounded growth.
Limits at infinity
As x → ∞, what does f(x) approach?
lim (x→∞) 1/x = 0 → the horizontal asymptote y = 0.
Why limits matter
Limits let calculus define:
- Instantaneous speed (the derivative)
- Exact areas under curves (the integral)
- Continuity
Worked examples
Example 1: lim (x→4) (x² − 1)? Substitute: 15.
Example 2: From a table, f(x) → 5 from the left and 5 from the right at x = 2. Limit? 5.
Example 3: lim (x→∞) 3/x? 0.
Common mistakes
- Thinking the limit must equal f(a).
- Checking only one side.
- Treating ∞ as a number.
Try it yourself
- lim (x→2) (3x − 4)
- Use a table to estimate lim (x→1) (x² − 1)/(x − 1).
- If the left limit is 6 and the right limit is 6, what is the limit?
- lim (x→∞) 5/x²?
Answers: 1) 2 2) 2 3) 6 4) 0
Quick recap
- A limit describes what f(x) approaches.
- The function doesn’t have to be defined at the point.
- Two-sided limits need matching one-sided limits; otherwise the limit does not exist.