Calculus: Analytic Geometry and Calculus, with Vectors

3.2 Limits 127

such that

(3.272) lim (1 + x) If- = e.

X-O

Anybody can collect a little evidence in support of this assertion by mak-

ing calculations when x has such values as ±J, ±j-, ± 4, and ±-L, but

it is not so easy to prove the assertion. In fact we must have very sub-

stantial information about limits before we can, in Chapter 9, define

functions having values ex and, when x > 0, log x. Meanwhile, many of

the problems that confront us will be solved very quickly and easily with

the aid of the following fundamental theorems. We call them limit

theorems, but they are nothing but basic theorems in the theory of

approximation.

Theorem 3.281 If

lim f(x) = L1, limf(x) = L2,

then L2 = L1.

Theorem 3.282 If b is a constant, then

Theorem 3.283

lim b = b.

lim x = a.

x-a

Theorem 3.284 If c is a constant, then

lim cf(x) = c lim f (x)

x-ia z-sa

provided the limit on the right exists.

Theorem 3.285 The formulas

lim [f(x) + g(x)] = lim f(x) + lim g(x)

x- a x-+a

lim [f(x)g(x)] = [lim f(x)][lim g(x)]

z-a

/

2a x-+a

f(x) - _

llima f(x)

lim

,ag(x) lim g(x)z-4a

are valid provided the limits on the right exist and, in the case of the last

formula, lim g(x) 0.

x- a

Theorem 3.286 If

lim f (x) = L

z a

then

limlf(x) - LI = 0

and conversely.