Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 2 REVIEW ¤ 81
19. Let t =1/x. Thenasx → 0 + , t →∞,and lim
x→0 + tan−1 (1/x) = lim
t→∞
tan −1 t = π 2 .
21. From the graph of y = cos 2 x /x 2 , it appears that y =0is the horizontal
asymptote and x =0is the vertical asymptote. Now 0 ≤ (cos x) 2 ≤ 1
0
x ≤ cos2 x
≤ 1 ⇒ 0 ≤ cos2 x
≤ 1 .But lim
2 x 2 x 2 x 2 x 0=0and
2 x→±∞
lim
x→±∞
1
x
2
=0, so by the Squeeze Theorem, lim
x→±∞
cos 2 x
x 2 =0.
⇒
cos 2 x
Thus, y =0is the horizontal asymptote. lim = ∞ because cos 2 x → 1 and x 2 → 0 as x → 0, sox =0is the
x→0 x 2
vertical asymptote.
23. Since 2x − 1 ≤ f(x) ≤ x 2 for 0 0 such that if 0 < |x − 2|