Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

156 0 CHAPTER 13 VECTOR FUNCTIONS
(d) lim u (t) x ·lim v (t) = ( lim u1 {t), l.im u2(t), lim u3(t)) x ( lim v1(t), lim v2(t), lim va(t))
t-a t-a t- a t--+a t.-a t-+o. t-+a t-ta.
= ([lim u2(t)] [lim v3(t)] - [lim us(t)] [lim v2(t)] ,
t-+a t-+a t-+n t -+a.
[lim ua(t)] [lim v1(t)] -
[lim u1(t)] [tim vs(t)],
t-+a t-+a t -+a t-+a
[p~ u1(t)) [~~ v2(t)) - [l~ u 2(t)] [E~ vl(t) ])
= (J~ [u2(t)v3(t)- u3(t)v2(t)] , E~ [ua(t)v1(t) - u1(t)va(t)] ,
~~ [u1(t)v2(t)- u2(t)v1(t)])
= lim (·u2(t)va(t)- ua(t)v2(t), U3 (t) v1 (t) - u1(t)v3(t), u1(t)v2(t)- tt2(t)v1 (t))
t-+a ·
= lim (u (t) x v (t)]
t-
51 . Let r (t) = (! (t), g (t), h (t)) and b = {b1, b2, ba) . If lim r (t) = b, then lim r (t) exists, so by (1),
t-a.
t-+a
b = lim r (t) ~ ( lim f(t), lim g(t), lim h(t)). By the definition of equal vectors we have lim f(t)·= b1, lim g(t) = b2
t-+a t-a t-a t -+a t-a t -+a
and lim h(t) = b 3 • But these are limits of real-valued functions, so by the definition of limits, for every c > 0 there exists
t-Hl
0, 02 > 0, oa > 0 so that ifO < It - a l < 01 then lf(t) - b1 l < c/3, ifO < it- al < 02 then lg(t) - b2l < c/ 3, and
ifO < It- al 0 there exists o > 0 such that if 0 < It - a l < o then
lr.(t)- b l ::; !f(t) - b1 l + lg(t) - b2 l + lh(t) - b3 l 0, there exists o > 0 such
tharifO l(f(t)- b1 ,g(t) - b2,h(t) - ba) l < c *>
J [f(t) - b!) 2 + [g(t)- b2j2 + [h(t) - baj2 < c ¢:> (J(t) - b 1 ] 2 + [g(t)- b2] 2 + [h(t) - baf < c 2 . But each term
on the left side of the last inequality is positive, so if 0 < It - al < o, then [! ( t) - b1] 2 < c 2 , [g( t) - b2] 2 < c 2 and
[h(t) - b3] 2 < c 2 or, taking the square root of both sides in each ofthe above, lf(t) - b1 l < c, lg(t) - b2 l < . c and
lh(t) - b3l