200 Chapter 4 • Limits of Functions

Theorem 4.2.23 (Change of Variables in Limits) Suppose lim g(x) =
x-+xo·
uo and lim f(u) = L , where x 0 and u 0 are c.luster points of V(g) and V(J),
U-+Uo
respectively, and g( x) E V(J) - { uo} for all x E V(g) in some deleted neigh-
borhood of xo. Then

lim f (g(x)) = lim f(u) = L.

x-+xo u-+uo

...-----f g L

f(u~

f(g(x)) = f(u)

Ni,(x 0 ) n 'D(g) g V(f)

X ~ XQ ------ U ~· UQ

g(x) = u

[as x ~ xa. g(x) ~ ua.] [as u ~ u 0 ,J(u) ~ L.]

Figure 4.6

Proof. Suppose lim g(x) = uo and lim f(u) = L , where xo and uo are
x-+xo u-+uo
cluster points of V(g) and V(f), respectively, and 3 6 > 0 3 'llx E N8( x 0 ) nV(g),
g(x) E V(f) - {uo}.
Let c > 0. Since lim f(u) = L, 361 > 0 3 Vu E V(f),
U-+Uo

0 < lu -uol < 61 =? lf(u) - LI < c.

Since lim g(x) = uo, 362 > 0 3 'llx E V(g),

X-+Xo

0 < Ix - xol < 62 =? lg(x) - uol < 61.

Choose 63 =min{ 6, 62}. Then Vx E V(f(g)J, x E V(g) and

0 < Ix -xol < 63 =? 0 <Ix -xol < 6 and 0 <::Ix -xol < 62

=? g(x) E V(J) - uo, and lg(x) - uol < 61

=? g(x) E V(f), and 0 < lg(x) - uol < 61

=? lf(g(x)) - LI < c.

Therefore, X-+Xo lim f (g(x)) = L = U-+Uo lim f(u). •