Solução_Calculo_Stewart_6e

F.
66 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
65. (a) 1/x 2 < 0.0001 ⇔ x 2 > 1/0.0001 = 10 000 ⇔ x>100 (x >0)
(b) If ε>0 is given, then 1/x 2 1/ε ⇔ x>1/ √ ε.LetN =1/ √ ε.
Then x>N ⇒ x> √ 1 ⇒
ε 1
x − 0 = 1 1
M.Nowe x >M ⇔ x>ln M, sotake
N =max(1, ln M). (ThisensuresthatN>0.) Then x>N=max(1, ln M) ⇒ e x > max(e, M) ≥ M,
so lim
x→∞ ex = ∞.
71. Suppose that lim f(x) =L. Then for every ε>0 there is a corresponding positive number N such that |f(x) − L| N.Ift =1/x,thenx>N ⇔ 0 < 1/x < 1/N ⇔ 0 0 (namely 1/N ) such that |f(1/t) − L|