126 3 Real Analysis
Definition.Forx 0 an accumulation point of the domain of a functionf, we say that
limx→x 0 f(x)=Lif for every neighborhoodVofL, there is a neighborhoodUofx 0
such thatf(U)⊂V.
This definition is, however, seldom used in applications. Instead, it is more customary
to use operations with limits, the squeezing principle (iff(x)≤g(x)≤h(x)for allxand
limx→x 0 f(x)=limx→x 0 h(x)=L, then limx→x 0 g(x)=L), continuity, or L’Hôpital’s
theorem, to be discussed later.
Example.Compute
lim
x→∞(√
x+√
x+√
x−√
x)
.
Solution.The usual algorithm is to multiply and divide by the conjugate to obtain
lim
x→∞(√
x+√
x+√
x−√
x)
=lim
x→∞x+√
x+√
x−x
√
x+√
x+√
x+√
x=lim
x→∞√
x+√
x
√
x+√
x+√
x+√
x=lim
x→∞√
1 +
√
1
x
√
1 +√
1
x+√
1
x^3 +^1=
1
2
.
And now an example of type 1∞.Example.Leta 1 ,a 2 ,...,anbe positive real numbers. Prove that
lim
x→ 0(
ax 1 +ax 2 +···+axn
n)^1 x
=n√
a 1 a 2 ···an.Solution.First, note that
lim
x→ 0ax− 1
x
=lna.Indeed, the left-hand side can be recognized as the derivative of the exponential at 0. Or
to avoid a logical vicious circle, we can argue as follows: letax= 1 +t, witht→0.
Thenx=ln(ln^1 +at), and the limit becomes
lim
t→ 0tlna
ln( 1 +t)=lim
t→ 0lna
ln( 1 +t)t=
lna
lne=lna.