Answers & Hints for Selected Exercises 655(e) f(x 0 ) = 2 and f(xci) = 3, so lim f(x) does not exist.
X-tXQ
(g) f(x 0 ) = 0 and f(xci) = 1, so lim f(x) does not exist.
X-tXQ- If 3 sequence {xn} in D(f) n (-oo,xo) 3 Xn --+ Xo but {f(xn)}-;.. L then
f does not have limit L from the left at x 0. If 3 sequence { xn} in D(f) n
(xo, +oo) 3 Xn--+ xo but {f(xn)} f> L then f does not have limit L from the
right at xo. - If lim f(x) = L -:/:-0, then f is bounded away from 0 on some interval of
x--+xQ
the form (xo - 6, xo). If lim f(x) = L-:/:-0 , then f is bounded away from 0 on
X-4Xci
some interval of the form ( xo, xo + 6). - Since AnB = 0, the statement, Vx E AUE, 0 < lx-xol < 6 =? lh(x)-LI <
c lS. eqmva. 1 en tt o {VxEA,O<lx - xol<6=?lf(x)-Ll<c,and}.
Vx EB, 0 < Ix - xol < 6 =? lg(x) - LI < c
EXERCISE SET 4.4-A- (a) Let M > 1. Choose 6 = it. Then 0 < Ix I < 6 =? 0 < x^2 < Ix I < it =?
~>M.
(c) Let M > 0. Choose 6 = JM· Then 0 < Ix+ ll < 6 =? (x.;l)2 > p = M.( e) Let M > 0. Choose 6 = min { ~, ~ }. Then 0 < Ix - 2 I < 6 =?
Ix -^21 < .! and Ix -^21 < -(^1) - =? x - 1 > .! and Ix - 21 < - (^1) -
2 ,/2M 2 ,/2M
=} (x-2)2 x-1 > 2 ·^1 2M = M =} (x-2)2 1-x < - M ·
- (a) lim f(x) = + oo ¢::? VM > 0, 3 6 > 0 3 xo - 6 < x < xo =? f(x) > M.
x--+xQ
(b) lim f(x) = -oo ¢::? VM > 0, 3 6 > 0 3 xo -6 < x < xo =? f(x) < -M.
x-+xQ
(c) lim f(x) = +oo ¢::? VM > 0, 3 6 > 0 3 xo < x < xo + 6 =? f(x) > M.
x-+xci
(d) lim f(x)=-oo¢:?VM>0,36>03xo<x<xo+6=?f(x)<-M.
x--+xci - (a) lim f(x) = +oo ¢::? V {xn} in D(f)n(-oo, xo) 3 Xn--+ xo, f(xn)--+ +oo.
x--+xQ
(b) lim J(x) = -oo ¢::? V {xn} in D(f)n(xo, +oo) 3 Xn--+ Xo, f(xn)--+ -oo.
X-tXQ - (a) lim f(x) = +oo ¢::? VM > 0, 3 6 > 0 3 0 < Ix - xol < 6 =? f(x) > M
X-tXQ
¢::? VM > O, 3 6 > 0 3 { xo - 6 < x < xo =? f(x) > M, and}
xo < x < xo + 6 =? f(x) > M. - Let M > 0. Then 3 61 > 0 3 0 <Ix - xol < 61 =? f(x) > ,/M,