1549901369-Elements_of_Real_Analysis__Denlinger_

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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). •
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