Pillow English Listening

• 511 Limit
ᐥLet f:A￫ℝ be a function whose domain A contains the interval [a,∞) for some fixed real value of a. Then f(x) tends to L as x tends to infinity, if for every ε>0 there exists a real number X such that x>X ⇒ |f(x)−L|<ε. ┆｢f(x) Ƚx￫∞｣ = L┆ ᐥ
9732#9733 SIBLINGS CHILDREN 9733

• 512 Limit
ᐥf(x) tends to the limit L as x tends to a if and only if for every ε>0 there exists a δ>0 such that 0<|x−a|<δ ⇒ |f(x)−L|<ε. ┆｢f(x) Ƚx￫a｣ = L┆ ᐥ
9732#9734 SIBLINGS CHILDREN 9734

• 513 Rules
ᐥSuppose that ｢f(x) Ƚx￫a｣=L and ｢g(x) Ƚx￫a｣=M. Then the following rules apply. 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 ...
9732#9735 SIBLINGS CHILDREN 9735

• 522 Continuous
f is continuous at c if and only if for every ε>0 there exists δ>0 such that |x−c|<δ ⇒ |f(x)−f(c)|<ε. ┆｢f(x) Ƚx￫c｣ = f(c)┆
9732#9756 SIBLINGS CHILDREN 9756

◌◌◌ Practice
9732#10488 SIBLINGS CHILDREN 10488