1549901369-Elements_of_Real_Analysis__Denlinger_

4.3 One-Sided Limits 207

Proof. Suppose x 0 is a cluster point of V(f) n(-oo, x 0 ), and a cluster

point of'D(J) n(xo,oo).

Part 1 (==>): Suppose lim f(x) = L. Let c; > 0. Since lim f(x) = L,

X--+Xo X-+Xo

38 > 0 3 'r:/x E V(J), 0 <Ix - xol < 8 ==> lf(x) -LI< c;. Then 'r:/x E V(J),

xo - 8 < x < xo * 0 < Ix - xol < 8 ==> lf(x) - LI< c;; and

xo < x < Xo + 8 * 0 < Ix - xol < 8 * lf(x) - LI< c;.

Therefore, lim f(x) =Land lim f(x) = L.
x--+x 0 x--+xci
Part 2 (<;:=):Suppose lim f(x) =Land lim f(x) = L. Let c; > 0. Since
x--+x 0 x--+xci
lim_ f(x) = L, 3 81 > 0 3 'r:/x E 'D(J), xo - 81 < x < xo ==> lf(x) - LI < c;.
X--+Xo
Since lim f(x) = L, 3 82 > 0 3 'r:/x E 'D(J), Xo < x < Xo +82 ==> lf(x)-LI < c.
x--+xri
Choose 8 = min{8 1 ,82}. Then, 'r:/x E 'D(J), 0 < Ix - xol < 8 ==>either xo -8 <
x < xo or xo < x < Xo + 8. In either of these cases, lf(x) - LI < c;. Thus,
'r:/x E 'D(J), 0 < Ix - xol < 8 ==> lf(x) -LI < E:. Therefore, lim f(x) = L. •
X--+Xo

Since the hypothesis of the previous theorem is rather complicated, we
restate the theorem with a slightly simpler hypothesis. The theorem is often
applicable in this form.

Corollary 4.3.9 If V(J) contains a deleted nbd. of xo, then lim f(x) = L {::}

X-+Xo

both lim -f(x) = L and lim + f(x) = L.

X--+Xo X--+Xo

Theorem 4.3.8 and its corollary are often useful in proving that lim f(x)
x --+xo
does not exist, as shown in the following example.

. lx-21.
Example 4.3.10 Prove that hm --does not exist.
X--+2 X - 2
.. lx-21.
Solution: In Example 4.3.2, we saw that hm --
2

• = -1 and m Ex-
  x--+2- X -

ample 4.3.4 we saw that lim Ix -
2
1 = l. Since these two one-sided limits at
X--+2+ X - 2

. lx-21.
2 are not equal, Corollary 4.3.9 above tells us that hm --
2
does not exist.
X--+ 2 X -
D