❶ sum rule
｢f(x)+g(x) Ƚx￫a｣ = L+M
❷ product rule
｢f(x)·g(x) Ƚx￫a｣ = L·M
❸ quotient rule
｢f(x)/g(x) Ƚx￫a｣ = L/M provided that M≠0ᐥ
Proof of ❶
∀ε>0 ∃δ>0 such that 0<|x−a|<δ ⇒ ∣f(x)−L∣ < ε/2
∀ε>0 ∃δ>0 such that 0<|x−a|<δ ⇒ ∣g(x)−M∣ < ε/2
Then
∀ε>0 ∃δ>0 such that 0<|x−a|<δ ⇒
∣(f(x)+g(x))−(L+M)∣
┆triangle inequality┆
≤ ∣f(x)−L∣+∣g(x)−M∣
┆by conditionals┆
< ε/2+ε/2
= ε
