Proofs of basic theorems on limits 717Theorem C
(8) lim x = a.
This theorem tells us that if f(x) = x, then(9) lim f(x) = a.
To prove the theorem, we observe that ife > 0 and we set a = e, then
Ix - al < c whenever 0 < Ix- al < S.
Theorem D
If c is a constant, then(10) lim cf (x) = c lim f (x)
x-+a x-4aprovided the limit on the right exists.
To prove this theorem, let lim f(x) = L. In case c = 0, the result is
a consequence of the fact that both sides of (10) are 0. In case c 5,16 0,
let e > 0 and choose a positive number S such that
I f(x) - LI < I;I
when Ixi a and Ix - al < S. Then
(12) Icf(x) - c lim f(x) I < ewhen x 0 a and Ix - al < S. This proves (10).
Theorem E
The formulas(13) lim [f(x) + g(x)] = lim f(x) + lim g(x)
x-a z-+a
(14) lim [f(x)g(x)] = [lim f(x)][lim g(x)]
x-a Xya xyaf(x)__urn
af(x)
(15) lma g(x)
lim g(x)are valid provided the limits on the right exist and, in the case of the last
formula, lim g(x) 0 0.
x--a
To prove these results, let(16) lim f(x) = L, lim g(x) = M.