The concept that makes calculus possible — understanding what happens as we approach, not just arrive.
The Intuition Behind Limits
A limit describes the value that a function approaches as its input approaches a particular value. We don't care what happens at the point — only what happens near it.
lim (x→c) f(x) = L
"As x gets closer and closer to c, f(x) gets closer and closer to L"
If f is continuous at c (no holes, jumps, or asymptotes), then lim(x→c) f(x) = f(c). All polynomials are continuous everywhere, so you can always plug in directly.
2. Algebraic Simplification
Factor, cancel, rationalize — use factoring techniques to eliminate the 0/0 form.
3. L'Hôpital's Rule (Preview)
If lim f(x)/g(x) gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x). This requires derivatives, covered in the next lesson.
4. Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) near c, and lim g(x) = lim h(x) = L, then lim f(x) = L.
This cannot be proved by algebra alone — it requires the Squeeze Theorem with geometric arguments from the unit circle. This limit is the foundation of all of trigonometric calculus.
Sometimes the limit depends on which direction you approach from:
lim (x→c⁺) f(x) = limit from the right
lim (x→c⁻) f(x) = limit from the left
The two-sided limit exists only if both one-sided limits exist and are equal. One-sided limits are essential for understanding piecewise functions and step functions in probability.
Limits at Infinity
What happens to f(x) as x → ∞? This determines the end behavior of functions and the existence of horizontal asymptotes.
Key results:
lim (x→∞) 1/xⁿ = 0 (for n > 0)
lim (x→∞) eˣ = ∞
lim (x→∞) e⁻ˣ = 0
lim (x→∞) ln(x) = ∞ (but grows slower than any positive power of x)
If f is continuous on [a, b] and N is between f(a) and f(b), then there exists some c in (a, b) where f(c) = N. This guarantees that equations have solutions and is used in numerical methods for differential equations.
Continuity connects deeply to topology in geometry. A continuous function is one that "preserves nearness" — nearby inputs map to nearby outputs. This intuition leads to coordinate geometry and abstract topology.
The Epsilon-Delta Definition (Advanced)
The rigorous definition of a limit, formalized by Weierstrass in the 19th century:
lim (x→c) f(x) = L means:
For every ε > 0, there exists δ > 0 such that
if 0 < |x − c| < δ, then |f(x) − L| < ε
In plain English: no matter how small a tolerance ε you demand for the output, I can find a tolerance δ for the input that guarantees the output is within ε of L.
This definition doesn't use the word "approach" — it's purely about inequalities and existence of numbers. This level of rigor resolved centuries of confusion about infinitesimals and made calculus logically watertight. It's a beautiful example of how number theory and analysis interact with calculus.