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Infinite Limits and Limits at Infinity

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2.3. Infinite limits and limits at infinity
2.3 Infinite limits and limits at infinity
Consider the function f (x) = x1 . What happens to f (x) as x → 0?
1
x
f (x)
x
f (x)
.5
1
2
-1
.1
.001
10
1000
- .5
−1
−2
-.1
-.001
−10
−1000
.0000001
1000000
-.0000001
−1000000
4
2
-4
2
-2
4
-2
-4
We invent new notation to describe this situation. We say
lim+
x→0
1
=∞
x
and
lim−
x→0
1
= −∞.
x
43
2.3. Infinite limits and limits at infinity
Formally:
• to say lim+ f (x) = ∞ means that as x gets closer and closer to a from the right,
x→a
the numbers f (x) grow without bound.
a
• to say lim+ f (x) = −∞ means that as x gets closer and closer to a from the
x→a
right, the numbers f (x) become more and more negative without bound.
a
• to say lim− f (x) = ∞ means that as x gets closer and closer to a from the left,
x→a
the numbers f (x) grow without bound.
a
• to say lim− f (x) = −∞ means that as x gets closer and closer to a from the
x→a
left, the numbers f (x) become more and more negative without bound.
a
All these situations are called infinite limits.
infinite limit is as follows:
The graphical description of an
44
2.3. Infinite limits and limits at infinity
Definition 2.3 If lim+ f (x) = ±∞ or lim− f (x) = ±∞, we say the vertical line
x→a
x→a
x = a is a vertical asymptote (VA) for f (x).
Example: x = 0 is a VA for f (x) = x1 .
NOTE: ∞ is not a number. It is only a symbol. However, in the context of
limits, ∞ can be manipulated in some ways as if it was a number (we’ll see how in
Chapter 3). For now you should remember these facts:
lim+
x→0
1
=∞
x
lim−
x→0
1
= −∞
x
One infinite limit to memorize:
lim ln x = −∞
x→0+
1
1
e
Other infinite limits are computed using techniques we will study later, using
some rules of arithmetic with ∞.
45
2.3. Infinite limits and limits at infinity
Limits at infinity
We want to consider the values of f (x) when x gets larger and larger without
bound. For example, suppose f (x) = x1 :
x
f (x)
1
10
1
10
1
10000
10100
1010000
1
10000
1
10100
1
1010000
We say x→∞
lim f (x) = L if
• (heuristically) when x grows without bound, f (x) approaches L.
• (graphically) the graph of f looks like
L
L
or
We say lim f (x) = L if
x→−∞
• (heuristically) when x becomes more and more negative without bound, f (x)
approaches L.
• (graphically) the graph of f looks like
L
or
L
Definition 2.4 If x→∞
lim f (x) = L or lim f (x) = L, we say the horizontal line y = L
x→−∞
is a horizontal asymptote (HA) for f (x).
46
2.3. Infinite limits and limits at infinity
E XAMPLE
Consider the following graph of some unknown function f :
8
7
6
5
4
3
2
1
-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
-2
-3
-4
-5
-6
Based on this graph, find the following:
1. lim f (x)
x→∞
2. lim f (x)
x→−∞
3.
4.
lim f (x)
x→−3+
lim f (x)
x→−3−
5. lim f (x)
x→−3
6. lim+ f (x)
x→3
7. lim− f (x)
x→3
8. lim f (x)
x→3
9. the equation(s) of any vertical asymptote(s) of f
10. the equation(s) of any horizontal asymptote(s) of f
47
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