Limits & Continuity
The limit is the foundational concept of calculus. It asks: as x approaches some value a, what value does f(x) approach?
lim[x→a] f(x) = L
means: for every ε > 0, there exists δ > 0 such that
if 0 < |x - a| < δ, then |f(x) - L| < ε
You don't need the formal ε-δ definition to compute limits — but understanding it is essential for theoretical CS (computability, numerical analysis).
Limit Laws
| Law | Formula |
|---|
| Sum | lim(f+g) = lim f + lim g |
| Product | lim(f·g) = lim f · lim g |
| Quotient | lim(f/g) = lim f / lim g (if lim g ≠ 0) |
| Power | lim f(x)ⁿ = (lim f(x))ⁿ |
Worked Example — Evaluating a Limit
Find: lim[x→2] (x² − 4)/(x − 2)
Direct substitution gives 0/0 — indeterminate form. Factor the numerator.
(x² − 4) = (x+2)(x−2), so (x²−4)/(x−2) = (x+2)(x−2)/(x−2)
Cancel (x−2): simplifies to (x+2) for x ≠ 2
Now substitute x=2: lim = 2+2 = 4 ✓
Continuity
f is continuous at a if: (1) f(a) is defined, (2) lim[x→a] f(x) exists, (3) lim[x→a] f(x) = f(a).
Knowledge Check
lim[x→2] (x²−4)/(x−2) equals
A function f is continuous at a if
lim[x→0] sin(x)/x equals
Which technique resolves a 0/0 indeterminate form?
The limit laws allow us to