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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
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136 Chapter 1 • Limits and Continuity
Teaching Tip
Students may not find it intuitive that the
x
lim e = 0.
x→−∞
Consider showing them the following steps:
lim e
x
x→−∞
= lim 1 Rewritethe expression.
x→−∞
e
− x
= 0 As x approaches negative
infinity,the limit approaches
zero.
y
4
2
(0, 1)
2
Figure 59 f (x) = e x
2
x
Other Infinite Limits at Infinity
We have seen that all polynomial functions have infinite limits at infinity, and some
rational functions have infinite limits at infinity. Other functions also have an infinite
limit at infinity.
For example, consider the function f (x) = e x .
The graph of the exponential function, shown in Figure 59, suggests that
lim
x→−∞ ex = 0
lim
x→∞ ex =∞
These limits are supported by the information in Table 15.
TABLE 15
x −1 −5 −10 −20 x approaches −∞
f (x) = e x 0.36788 0.00674 0.00005 −2 × 10 −9 f (x) approaches 0
x 1 5 10 20 x approaches ∞
f (x) = e x e ≈ 2.71828 148.41 22,026 4.85 × 10 8 f (x) becomes unbounded
y
2
EXAMPLE 9
Find lim ln x.
x→∞
Finding the Limit at Infinity of g(x) =ln x
Solution Table 16 and the graph of g(x) = ln x in Figure 60 suggest that g(x) = ln x
has an infinite limit at infinity. That is,
2 4
x
lim ln x =∞
x→∞
2
4
Figure 60 g(x) = ln x
y
4
TABLE 16
x e 10 e 100 e 1000 e 10,000 e 100,000 → x becomes unbounded
g(x) = ln x 10 100 1000 10,000 100,000 → g(x) becomes unbounded
Now let’s compare the graph of f (x) = e x in Figure 59, the graph of g(x) = ln x
in Figure 60, and the graph of h(x) = x 2 in Figure 61. As x becomes unbounded in
the positive direction, all three of the functions increase without bound. But in Table 17,
f (x) = e x approaches infinity more rapidly than h(x) = x 2 , which approaches infinity
more rapidly than g(x) = ln x.
■
24
22
2
21
2
4
x
TABLE 17
x 10 50 100 1000 10,000
f (x) = e x 22,026 5.185 × 10 21 2.688 × 10 43 e 1000 e 10,000
h(x) = x 2 100 2500 = 2.5 × 10 3 1.0 × 10 4 1.0 × 10 6 1.0 × 10 8
g(x) = ln x 2.303 3.912 4.605 6.908 9.210
Figure 61 h(x) = x 2
EXAMPLE 10
Application: Decomposition of Salt in Water
Salt (NaCl) dissolves in water into sodium (Na + ) ions and chloride (Cl − ) ions according
to the law of uninhibited decay
A(t) = A 0 e kt
where A = A(t) is the amount (in kilograms) of undissolved salt present at time t (in
hours), A 0 is the original amount of undissolved salt, and k is a negative number that
represents the rate of dissolution.
136
Chapter 1 • Limits and Continuity
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