This preview shows page 1. Sign up to view the full content.
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
²
·
±
lim
k
→∞
1
k
²
= 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
²
·
±
lim
k
→∞
b
n
²
(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
²
·
±
lim
k
→∞
(

1)
k
²
.
But this expression doesn’t make sense because the sequence
³
(

1)
k
´
∞
k
=1
doesn’t have a limit!
2. Consider
1 = lim
k
→∞
1 = lim
k
→∞
k
·
1
k
=
±
lim
k
→∞
k
²
·
±
lim
k
→∞
1
k
²
=
±
lim
k
→∞
k
²
·
0
.
This expression doesn’t make sense either, since the sequence
{
k
}
∞
k
=1
doesn’t have a limit.
These examples should help us see what kind of care we need to take when using principle (2). Namely,
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 Vuletic,M
 Calculus, Limits

Click to edit the document details