Example 2.
Show that lim
x
→∞
1
x
2
= 0.
Solution.
Fix
ε >
0.
We need
c
so that if
x > c
then 1
/x
2
< ε
.
Choosing
c
= 1
/
√
ε
works.
Basic limit laws
One can show that the basic limit laws hold for limits at
±∞
.
So if lim
x
→∞
f
(
x
) =
A
,
lim
x
→∞
g
(
x
) =
B
where
A, B
are real numbers (i.e. not infinite!) then
f
(
x
) +
g
(
x
)
, f
(
x
)

g
(
x
)
, f
(
x
)
g
(
x
) will also have limits at
∞
(
A
+
B, A

B
and
AB
), and if
B
6
= 0 then
f
(
x
)
/g
(
x
)
will have a limit
A/B
. (Similar statements hold at
∞
.) You can also use the Squeezing
Principle, this is especially useful if you can estimate complicated functions with simpler
ones.
If you want to use the limit laws for infinite limits then you have to be a bit more careful:
remember that
∞
and
∞
are not real numbers. However you can still ‘pretend’ that certain
operations can be carried out:
‘
∞
+
∞
=
∞
’, ‘
∞ × ∞
=
∞
’, ‘
c
× ∞
=
∞
’ if
c >
0 and ‘
c
× ∞
=
∞
’ if
c <
0, ‘
c
∞
= 0’,
c
+
∞
=
∞
’ etc. You should actually use these as rules for limits, not as the outcomes of
operations on
∞
. That means that you will have statements like this:
Assume that
lim
x
→
p
f
(
x
) =
∞
and
lim
x
→
p
g
(
x
) =
∞
.
Then
•
lim
x
→
p
(
f
(
x
) +
g
(
x
)) =
∞
1
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•
lim
x
→
p
f
(
x
)
g
(
x
) =
∞
•
lim
x
→
p
4
f
(
x
)
= 0.
Note that if
f
and
g
have
∞
limits at
p
then we cannot say anything about the limit of
f
(
x
)

g
(
x
) and
f
(
x
)
/g
(
x
) there. (It can be basically anything or it might not exist.)
Polynomials and rational functions
It is not hard to show that for any nonconstant polynomial
p
(
x
) the limit of
p
(
x
) will be
∞
or
∞
ad
x
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 Fall '08
 Staff
 Calculus, Limits, Limit, Fraction, Rational function, ∞

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