Chapter 1. Limits and Continuity
1.3. Limits Involving Infinity
Definition. Formal Definition of Limits at Infinity. (NOT IN
10TH EDITION!)
1. We say that f (x) has the limit L as x approaches infinity and we
write
lim f (x) = L
x→+∞
if, for every number > 0, there exists a corresponding number M
such that for all x
x>M
⇒
|f (x) − L| < .
2. We say that f (x) has the limit L as x approaches negative infinity
and we write
lim f (x) = L
x→−∞
if, for every number > 0, there exists a corresponding number N
such that for all x
x<N
⇒
|f (x) − L| < .
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Definition. Informal Definition of Limits Involving Infinity.
1. We say that f (x) has the limit L as x approaches infinity and write
lim f (x) = L
x→+∞
if, as x moves increasingly far from the origin in the positive direction,
f (x) gets arbitrarily close to L.
2. We say that f (x) has the limit L as x approaches negative infinity
and write
lim f (x) = L
x→−∞
if, as x moves increasingly far from the origin in the negative direction,
f (x) gets arbitrarily close to L.
Example. Example 1 page 112. Show that x→∞
lim
1
= 0.
x
Solution. Let > 0 be given. We must find a number M such that for
all
1
1
x>M ⇒
− 0 = < .
x
x
The implication will hold if M = 1/ or any larger positive number (see
1
the figure below). This proves x→∞
lim = 0. We can similarly prove that
x
1
= 0.
QED
lim
x→−∞ x
2
Figure 1.3.34, page 221 of 9th edition.
Theorem 7. Rules for Limits as x → ±∞.
If L, M , and k are real numbers and
lim f (x) = L
x→±∞
and
lim = M,
x→±∞
1. Sum Rule: lim (f (x) + g(x)) = L + M
x→±∞
2. Difference Rule: lim (f (x) − g(x)) = L − M
x→±∞
3. Product Rule: lim (f (x) · g(x)) = L · M
x→±∞
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then
4. Constant Multiple Rule: lim (k · f (x)) = k · L
x→±∞
f (x)
L
= , M = 0
x→±∞ g(x)
M
5. Quotient Rule: lim
6. Power Rule: If r and s are integers, s = 0, then
lim (f (x))r/s = Lr/s
x→±∞
provided that Lr/s is a real number AND L > 0.
Note. As in section 1.1, there is an error in the text with part 6 of
1
Theorem 7, as can be seen by considering lim √ which clearly does
x→−∞
x
not exist (but would be 0 by the use of Theorem 7, part 6 as stated in the
text, with f (x) = 1/x, r = 1 and s = 2).
Example. Page 122 number 10.
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Definition. Infinite Limits
1. We say that f (x) approaches infinity as x approaches x0, and we
write
lim f (x) = ∞,
x→x0
if for every positive real number B there exists a corresponding δ > 0
such that for all x
0 < |x − x0| < δ
⇒
f (x) > B.
2. We say that f (x) approaches negative infinity as x approaches x0,
and we write
lim f (x) = −∞,
x→x0
if for every negative real number −B there exists a corresponding
δ > 0 such that for all x
0 < |x − x0| < δ
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⇒
f (x) < −B.
Figure 1.3.39 and 1.3.40, page 119.
Note. Informally, x→x
lim f (x) = ∞ if f (x) can be made arbitrarily large by
0
making x sufficiently close to x0 (and similarly for f approaching negative
infinity). We can also define one-sided infinite limits in an analogous
manner.
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Definition. Horizontal and Vertical Asymptotes.
A line y = b is a horizontal asymptote of the graph of a function y = f (x)
if either
lim f (x) = b
lim f (x) = b.
or
x→∞
x→−∞
A line x = a is a vertical asymptote of the graph if either
lim f (x) = ±∞
x→a+
or
lim f (x) = ±∞.
x→a−
Example. Page 122 number 32.
Definition. End Behavior Model
The function g is
(a) a right end behavior model for f if and only if
lim
x→∞
f (x)
=1
g(x)
(b) a left end behavior model for f if and only if
f (x)
= 1.
x→−∞ g(x)
lim
Definition. If g(x) = mx + b where m = 0 is an end behavior model for
f , then f is said to have an oblique (or slant) asymptote of y = mx + b.
Example. Page 122 number 38.
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