A small warning about taking limits
Math 8 2009, Chris Lyons
Consider the following evaluation of a limit:
lim
k
→∞
1
k
2
= lim
k
→∞
1
k
·
1
k
=
lim
k
→∞
1
k
·
lim
k
→∞
1
k
= 0
·
0 = 0
.
(1)
Here’s the basic principle that we used in this calculation:
lim
k
→∞
a
n
b
n
=
lim
k
→∞
a
n
·
lim
k
→∞
b
n
(2)
Q
:
How correct is this principle?
Before answering this question, let’s look at two more examples which make us think that there’s a little
more going on in (2) than first meets the eye:
1. Consider
1 = lim
k
→∞
1 = lim
k
→∞
(

1)
2
k
= lim
k
→∞
[(

1)
k
·
(

1)
k
] =
lim
k
→∞
(

1)
k
·
lim
k
→∞
(

1)
k
.
But this expression doesn’t make sense because the sequence
(

1)
k
∞
k
=1
doesn’t have a limit!
2. Consider
1 = lim
k
→∞
1 = lim
k
→∞
k
·
1
k
=
lim
k
→∞
k
·
lim
k
→∞
1
k
=
lim
k
→∞
k
·
0
.
This expression doesn’t make sense either, since the sequence
{
k
}
∞
k
=1
doesn’t have a limit.
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 Fall '08
 Vuletic,M
 Calculus, Limits, lim, k→∞ k lim

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