Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

22 Introdnction to Analysis (C/t. /
Sec. 3. Lfmits
1. The limit of a sequence. The number a is the limit of a sequence
*! x lt .... X0, .... or
lim x n a,
n > oo
if for any e>0 there is a number N = N (e) such that
\xn a | < e when n> N.
Example
1. Show that
Solution. Form the difference
Evaluating
if
Urn 5L + 1.2. (1)
n -* rt-r 1
2* +1
1
the absolute value of this difference, we have:
-2 1
< e, (2)
n>-\ = N (e).
Thus, for every positive number there will be a number Af= 1 such
that for n > N we will have inequality (2) Consequently, the number 2 is
the limit of the sequence x n (2n-\- l)/(n-fl), hence, formula (1) is true.
2. The limit of a function. We say that a function / (x) -*- A as x -+ a
(A and a are numbers), or
lim f(x) = A,
x -a
if for every 8 > we have 6 = 6 () > such that