James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 11.1 Sequences 697
If you compare Definition 2 with Definition 2.6.7, you will see that the only difference
between lim n l ` a n − L and lim x l ` f sxd − L is that n is required to be an integer.
Thus we have the following theorem, which is illustrated by Figure 6.
3 Theorem If lim x l ` f sxd − L and f snd − a n when n is an integer, then
lim n l ` a n − L.
y
y=ƒ
L
FIGURE 6
0 1 2 3 4
x
In particular, since we know that lim x l ` s1yx r d − 0 when r . 0 (Theorem 2.6.5),
we have
4
lim
n l `
1
n r − 0 if r . 0
If a n becomes large as n becomes large, we use the notation lim n l ` a n − `. The following
precise definition is similar to Definition 2.6.9.
5 Definition lim n l ` a n − ` means that for every positive number M there is an
integer N such that
if n . N then a n . M
If lim n l ` a n − `, then the sequence ha n j is divergent but in a special way. We say
that ha n j diverges to `.
The Limit Laws given in Section 2.3 also hold for the limits of sequences and their
proofs are similar.
Limit Laws for Sequences
If ha n j and hb n j are convergent sequences and c is a constant, then
lim sa n 1 b n d − lim a n 1 lim b n
n l ` n l ` n l `
lim sa n 2 b n d − lim a n 2 lim b n
n l ` n l ` n l `
lim ca n − c lim a n
n l ` n l `
lim c − c
n l `
lim sa nb n d − lim
n l ` n l ` an ? lim
n l ` bn
lim
a a n
n n l `
lim −
n l ` b n lim
n l ` bn
if lim
n l ` b n ± 0
lim
n l `an p − Flim
n l `anG p if p . 0 and a n . 0
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