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Real Analysis
· Formula🧮 Formula Reference Sheet
Limit: lim x→a f(x) = L ⇔ ∀ε>0 ∃δ>0: 0<|x−a|<δ → |f(x)−L|<ε Continuity: f continuous at a iff lim f(x) = f(a) Derivative: f'(a) = lim h→0 (f(a+h)−f(a))/h Cauchy sequence: |aₙ − aₘ| → 0
⚡ Example per formula
Limit: lim x→a f(x) = L
↳lim x→2 (x²) = 4
Continuity: f continuous at a iff lim f(x) = f(a)
↳f(x) = x² is continuous everywhere
Derivative: f'(a) = lim h→0 (f(a+h)−f(a))/h
↳f(x)=x², f'(a) = 2a
Cauchy sequence: |aₙ − aₘ| → 0
↳1, 1.4, 1.41, 1.414, … → √2
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