Solução_Calculo_Stewart_6e

F.
142 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
andusingtheadditionpropertyoflimits for real-valued functions, we have that
lim u(t) + lim v(t)= lim u1(t)+lim v1(t), lim u2(t)+lim
t→a t→a t→a t→a t→a t→a
=
lim
t→a
v2(t), lim u3(t)+limv3(t)
t→a t→a
[u1(t)+v1(t)] , lim [u2(t)+v2(t)] , lim [u3(t)+v3(t)]
t→a t→a
=lim
t→a
hu 1 (t)+v 1 (t),u 2 (t)+v 2 (t),u 3 (t)+v 3 (t)i
=lim[u(t)+v(t)]
t→a
(b) lim cu(t) =limhcu 1 (t),cu 2 (t),cu 3 (t)i = lim cu
t→a t→a
1(t), lim cu 2 (t), lim
t→a t→a
= c lim u
t→a
1(t),clim u
t→a
2(t),clim u
t→a
3(t) = c lim
t→a
= c lim hu
t→a
1(t),u 2(t),u 3(t)i = c lim u(t)
t→a
(c) lim u(t) · lim v(t) = lim u1(t), lim u2(t), lim u3(t) ·
t→a t→a t→a t→a t→a
lim
t→a
= lim u1(t) lim v1(t) + lim u2(t) lim v2(t)
t→a t→a t→a t→a
=lim
t→a
u 1(t)v 1(t)+lim
t→a
u 2(t)v 2(t)+lim
t→a
u 3(t)v 3(t)
cu 3 (t)
t→a
u1(t), lim u2(t), lim u3(t)
t→a t→a
v1(t), lim v2(t), lim v3(t)
t→a t→a
+ lim
[using (1) backward]
u3(t) lim v3(t)
t→a t→a
=lim[u t→a 1(t)v 1(t)+u 2(t)v 2(t)+u 3(t)v 3(t)] = lim [u(t) · v(t)]
t→a
(d) lim u(t) × lim v(t) = lim u
t→a t→a
1(t), lim u 2 (t), lim u 3 (t) × lim v
t→a t→a t→a
1(t), lim v 2 (t), lim v 3 (t)
t→a t→a t→a
= lim u 2(t) lim v 3(t) − lim u 3(t) lim v 2(t) ,
t→a t→a t→a t→a
lim u 3(t) lim v 1(t) − lim u 1(t) lim v 3(t) ,
t→a t→a t→a t→a
lim u 1(t) lim v 2(t) − lim u 2(t) lim v 1(t)
t→a t→a t→a t→a
=
lim
t→a
[u2(t)v3(t) − u3(t)v2(t)] , lim [u3(t)v1(t) − u1(t)v3(t)] ,
t→a
lim [u1(t)v2(t) − u2(t)v1(t)]
t→a
=lim
t→a
hu 2(t)v 3(t) − u 3(t)v 2(t),u 3 (t) v 1(t) − u 1(t)v 3(t),u 1(t)v 2(t) − u 2(t)v 1(t)i
=lim
t→a
[u(t) × v(t)]
45. Let r(t) =hf (t) ,g(t) ,h(t)i and b = hb 1,b 2,b 3i. Iflim r(t) =b,thenlim r(t) exists,soby(1),
t→a t→a
b =limr(t) = lim f(t), lim g(t), lim h(t) .Bythedefinition of equal vectors we have lim f(t) =b
t→a t→a t→a t→a t→a 1, lim g(t) =b 2
t→a
and lim
t→a
h(t) =b 3. But these are limits of real-valued functions, so by the definition of limits, for every ε>0 there exists
δ 1 > 0, δ 2 > 0, δ 3 > 0 so that if 0 < |t − a|