SECTION 5.1 Quick Survey of Limits 247
Definition. Let f be a function defined on an interval of the form
a < x < b. We say that thelimit of f(x) is Las x approachesa
from the right, and write
lim
x→a+
f(x) = L,if for any real number > 0 , there is another real numberδ > 0 (which
in general depends on ) such that whenever 0 < x−a < δ then
|f(x)−L|< .
Similarly, one defines lim
x→a−
f(x) = L.Limits behave in a very reasonable manner, as indicated in the fol-
lowing theorem.
Theorem. Letf andgbe functions defined in a punctured neighbor-
hood ofa.^1 Assume that
xlim→af(x) = L, xlim→ag(x) = M.
Then,
xlim→a(f(x) +g(x)) = L+M, and xlim→af(x)g(x) = LM.Proof. Assume that δ >0 has been chosen so as to guarantee that
whenever 0<|x−a|< δ, then
|f(x)−L|<2
, and |g(x)−M|<2
.
Then,
|f(x) +g(x)−(L+M)|<|f(x)−L|+|g(x)−M|<2
+
2
=,
proving that limx→a(f(x) +g(x)) = L+M.
(^1) A “punctured” neighborhood of the real numberais simply a subset of the form{x∈ R| 0 <
|x−a|< d, for some positive real numberd.