→ ∞
or
x
→ ∞
. (Eventually the main term of the polynomial will dominate,
so you will need to find the limit of
cx
n
as
x
→ ±∞
.)
For rational functions the limits at infinity can also be a real number, this will happen if
the degree for the numerator is at most as big as the degree of the denominator.
Example 3.
Show that lim
x
→∞
x
2

3
x
+4
2
x
2
+4
x

5
=
1
2
.
Solution.
In order to find the limit at
∞
, it is enough to evaluate the function for large
values of
x
. If
x >
0 then
x
2

3
x
+4
2
x
2
+4
x

5
=
1

3
x
+
4
x
2
2+
4
x

5
x
2
whenever both sides make sense. But now the
numerator has a limit equal to 1 as
x
→ ∞
and the limit of the denominator as
x
→ ∞
is 2
so the limit of the ration is
1
2
.
One can also show that if the rational function
f
(
x
) =
p
(
x
)
q
(
x
)
is not defined at
x
=
c
(because
q
(
c
) = 0) then lim
x
→
c

f
(
x
) and lim
x
→
c
+
f
(
x
) will always exist (but might be infinite).
Example 4.
Show that lim
x
→
2

x
2
+2
x

2
=
∞
and lim
x
→
2
+
x
2
+2
x

2
=
∞
.
Solution.
lim
x
→
2
x
2
+ 2 = 4 and the onesided limits of
1
x

2
are
∞
from the left and
∞
from
the right. From this the statement follows.
Practice problems
1. Write out the definition of the following statements using inequalities:
(a) lim
x
→
2
+
g
(
x
) =
∞
(b) lim
x
→∞
h
(
x
) =
∞
2. Find the following limits (you may use the limit laws)
(a) lim
x
→
2
1
(
x

2)
2
(b) lim
x
→
4

x
4

x
(c) lim
x
→∞
1
x
3
3. Assume that lim
x
→
2

f
(
x
) =
∞
and lim
x
→
2

g
(
x
) = 5. Show (with ‘
ε

δ
’ proofs) that
(a) lim
x
→
2

f
(
x
)
2
=
∞
(b) lim
x
→
2

f
(
x
)
g
(
x
) =
∞
(c) lim
x
→
2

g
(
x
)

f
(
x
) =
∞
2
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 Fall '08
 Staff
 Calculus, Limits, Limit, Fraction, Rational function, ∞

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