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"
Example: lim (x→2) (x² − 4)/(x − 2)
Direct substitution gives 0/0 (indeterminate!)
Factor: (x−2)(x+2)/(x−2) = x + 2 (for x ≠ 2)
lim (x→2) (x + 2) = 4
The function isn't defined at x = 2, but the limit exists. This is the key insight of calculus.
Computing Limits
1. Direct Substitution
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.
If g(x) ≤ f(x) ≤ h(x) near c, and lim g(x) = lim h(x) = L, then lim f(x) = L.
Famous Example: lim (x→0) sin(x)/x = 1
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.
One-Sided Limits
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)
A function f is continuous at x = c if three conditions hold:
f(c) is defined
lim (x→c) f(x) exists
lim (x→c) f(x) = f(c)
The Intermediate Value Theorem (IVT)
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.