1549901369-Elements_of_Real_Analysis__Denlinger_

4.3 One-Sided Limits 203

4.3 One-Sided Limits

As you recall from calculus, "one-sided" limits frequently make sense in situa-

tions in which the ordinary limit does not exist.

Definition 4.3.1 (Limit from the Left) If x 0 is a cluster point of D(f) n

(-oo, xo), then the limit off from the left is L , written f(x 0 ) = lim f(x) = L,

x--+x()

if

I Ve:> 0, 38 > 0 3 Vx E D(f), Xo - 8 < x < xo::::? lf(x) -LI< c:. I

Notes on Definition 4.3.1:

(1) We shall never say that lim_ f(x) exists unless x 0 is a cluster point of

D(f) n ( -oo, xo).

(2) Even if xo E D(f), the value of f(xo) is irrelevant to the consideration
of whether lim f(x) = L. The condition "xo - 8 < x < xo" in Definition 4.3.l
x--+xQ
guarantees that when we consider whether lim f(x) = L, we are never letting
x--+x;J
X =Xo.

(3) If D(f) contains some interval of the form (xo - 1, xo), for some/> 0,
then Definition 4.3. l simplifies to:

lim f(x) = L if Ve:> 0, 3 8 > 0 3 xo - 8 < x < xo::::? lf(x) -LI < c:.

X--+XQ

There is actually a third quantifier here. The universal quantifier on x is un-
derstood to be present, even when left out in the interest of simpli city.

( 4) The following statements are interchangeable, and each one will find
use at one time or another:

(i) lim f(x) = L.

x--+xQ

(ii) f(x 0 ) = L.

(iii) f has limit L as x approaches Xo from the left.

(iv) f has left-hand limit L at xo.

(v) f has limit L from the left at xo.

(vi) f(x) ---7 Las x ---7 x 0.