CHAPTER 2. LIMITS
2.6
37
Limits at infinity
1
both graphically and numerically.
x→∞ x
Example 1. Find lim
1
Solution. Graphically, we look at the graph of , and track the y-values as
x
x moves to the right. The limit of these y-values, also called the horizontal
1
asymptote, is y = 0. Thus, lim = 0.
x→∞ x
graphics/one_over_x_HA-eps-converted-to.pdf
Numerically, we can imagine a table like this:
x
10
100
1000
10000
10, 000, 000
1
x
0.1
0.01
0.001
0.00001
0.000 000 1
The numbers in the right hand column are all getting closer and closer to 0.
Example 2. Find the horizontal asymptote of
Solution.
3x8 − 100x3 + 17.304
= lim
x→∞
x→∞ 7x8 − x + 100, 000
lim
1
x8
1
x8
3x8 − 100x3 + 17.304
.
7x8 − x + 100, 000
3x8 − 100x3 + 17.304
7x8 − x + 100, 000
3x8 x18 − 100x3 x18 + 17.304 x18
x→∞ 7x8 18 − x 18 + 100, 000 18
x
x
x
= lim
100
+ 17.304
x5
x8
100,000
1
+ x8
x7
100
+ 17.304
∞
∞
100,000
1
+
∞
∞
3−
x→∞ 7 −
= lim
=
3−
7−
divide top and bottom by x8
distribute
simplify
Limit laws
CHAPTER 2. LIMITS
38
=
3−0+0
3
=
7−0+0
7
Facts about limits at infinity
3x2 + x + 1/x
Example 3. Find lim �
x→∞
9x4 + 1/x
Solution. The biggest “simplified” power on the bottom is x2 , since we have
x4 inside of a square root. We divide the top and bottom by x2 . At one step we
√
√
1
have to move 2 inside of a square root. Recall that a2 b = a b, so to move
x
�
1√
1
something inside a square root, it comes in squared. Thus 2 a =
a
x
x4
1
(3x2 + x + 1/x)
x2
�
1
9x4 + 1/x
x2
3 + x1 + x13
lim �
x→∞
1
(9x4 + x1 )
x4
3x2 + x + 1/x
lim �
lim
x→∞
9x4 + 1/x x→∞
=
divide top and bottom by
distribute
simplify
1
3+ ∞
+ ∞13
= �
9 + ∞15
Facts of limits at infinity
3+0+0
3
= √
= =1
3
9+0
x2 − 7x
Example 4. Find lim √
.
x→∞
x5 + 1
Solution. We start by dividing the top and bottom by x5/2
1
(x2 − 7x)
x5/2
√
1
x→∞
x5 + 1
5/2
x
2−5/2
1−5/2
x
= lim �
x→∞
1
(x5 + 1)
(x5/2 )2
1
1/2
= lim �x
x→∞
− 7x
−
1
(x5
x5
7
x3/2
1
7
−∞
∞
=�
1
1+ ∞
distribute
1
x5/2
simplify
simplify more
+ 1)
1
7
1/2 − x3/2
= lim x�
x→∞
1 + x15
x4 = x2
1
and move it inside the square root
x2
3 + 1 + 13
= lim � x x
x→∞
9 + x15
lim
√
apply basic facts
CHAPTER 2. LIMITS
39
0−0
=√
=0
1+0
Example 5. Find lim cos(x/π).
x→∞
Solution. We look at the graph of f (x)
graphics/cos_of_x_over_pi-eps-converted-to.pdf
From this graph, it is clear that no horizontal asymptote exists, and thus the
limit does not exist.
lim cos(x/π) = DNE
x→∞
Example 6. Find lim (
x→∞
�
4x2 + x + 1 − 2x).
Solution. The trick here is to first rewrite the function as a fraction, and
then rationalize the numerator.
√
lim ( 4x2 + x + 1 − 2x)
force it into a fraction
x→∞
√
4x2 + x + 1 − 2x
= lim
rationalize numerator
x→∞
1
√
√
4x2 + x + 1 − 2x
4x2 + x + 1 + 2x
= lim
·√
finish rationalizing
x→∞
1
4x2 + x + 1 + 2x
(4x2 + x + 1) − (2x)2
= lim √
simplify
x→∞
4x2 + x + 1 + 2x
x+1
= lim √
divide top and bottom by x
2
x→∞
4x + x + 1 + 2x
1
x+1
1
= lim x1 √
distribute
x→∞
x
4x2 + x + 1 + 2x
x
1
(x + 1)
1
bring inside square root
= lim 1 √ x
2
x→∞ ( 4x + x + 1 + 2x)
x
x
CHAPTER 2. LIMITS
= lim �
x→∞
1+
1
(4x2
x2
40
1
x
distribute
+ x + 1) + 2
1 + x1
= lim �
x→∞
4 + x1 + x12 + 2
=�
=√
1+
4+
1
∞
1
∞
+
1
∞
1
inside square root
x2
apply basic facts
+2
1
1+0
=
4
4+0+0+2
Example 7. Find lim (−28x11 + 1000x10 + 1).
x→∞
Solution.
This function is a polynomial, and its graph will not have a
horizontal asymptote, nor will it oscillate up and down like sine and cosine.
Rather, the limit will either be +∞ or −∞, and the problem is to determine
which of these is the answer.
As before, we will factor out powers of x. As before, we focus on the highest
power of x.
lim (−28x11 + 1000x10 + 1)
x→∞
1
1
= lim x11 (−28 + 1000 + 11 )
x→∞
x x
= ∞(−28 + 0 + 0) = −∞
In fact, this example generalizes to a familiar rule: the end behavior of a
polynomial depends only on the leading term, i.e. the term with the largest
power of x.