Calculus: Analytic Geometry and Calculus, with Vectors

Proofs of basic theorems on limits 719

whenever 0 < Ix - at < S. Therefore,

(28)

1 -.1 i _ M - g(x) < e2 2e2

g(x) MR ' g(x)M , =(M/2)M

__

M2

when 0 < Ix - at < S. If we choose e2 such that 2e2/M2 < e, we will
have

(29)

1

g(x) lim g(x) i< e

x-4a

whenever 0 < Ix - at < S. This proves (15) and completes the proof

of Theorem E.

Theorem F

If

(30)

then

(31)

and conversely.

lim f(x) = L

x-a

liralf(x) - LI = 0

x-.a

The assertion (30) means that

1 I

to each positive number a there corre-

sponds a positive number S such that

(32) If(x) -LI <e

whenever 0 < Ix - at < S. The assertion (31) means that to each

positive number a there corresponds a positive number S such that

(33) I If(x) - LI - 01 < e

whenever 0 < Ix - at < S. Since the left members of (32) and (33)

are equal, each assertion implies the other.

Theorem G (sandwich theorem, or flyswatter theorem)

If, for some positive number p,

(34) g(x) < f(x) 5 h(x)

whena - p <x <aandwhena<x<a+p,andif

(35) lim g(x) = L, lim h(x) = L,

then

(36) lim f(x) = L.

x-+a

The primitive idea behind this theorem may be phrased as follows.

If two slices of bread (or two books) are near Minneapolis and if a slice