James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

112 Chapter 2 Limits and Derivatives
To summarize,
if 0 , | x 2 a | , then | f sxd 1 tsxd 2 sL 1 Md | , «
Thus, by the definition of a limit,
lim f f sxd 1 tsxdg − L 1 M
■
x l a
Infinite Limits
Infinite limits can also be defined in a precise way. The following is a precise version of
Definition 2.2.4.
y
M
y=M
6 Precise Definition of an Infinite Limit Let f be a function defined on some
open interval that contains the number a, except possibly at a itself. Then
lim f sxd − `
x l a
means that for every positive number M there is a positive number such that
if 0 , | x 2 a | , then f sxd . M
0 a
x
a-∂ a+∂
figure 10
This says that the values of f sxd can be made arbitrarily large (larger than any given
number M) by requiring x to be close enough to a (within a distance , where depends
on M, but with x ± a). A geometric illustration is shown in Figure 10.
Given any horizontal line y 5 M, we can find a number . 0 such that if we restrict
x to lie in the interval sa 2 , a 1 d but x ± a, then the curve y 5 f sxd lies above the
line y − M. You can see that if a larger M is chosen, then a smaller may be required.
1
ExamplE 5 Use Definition 6 to prove that lim
x l 0 x − `. 2
SOLUtion Let M be a given positive number. We want to find a number such that
if 0 , | x | , then 1yx 2 . M
But
1
x 2 . M &? x 2 , 1 M
&?
sx ,Î 1 2 M
&? | x | , 1
sM
So if we choose − 1ysM and 0 , | x | , − 1ysM , then 1yx 2 . M. This shows
that 1yx 2 l ` as x l 0.
n
y
a-∂
a+∂
Similarly, the following is a precise version of Definition 2.2.5. It is illustrated by
Figure 11.
a
0 x
N
y=N
FIGURE 11
7 Definition Let f be a function defined on some open interval that contains
the number a, except possibly at a itself. Then
lim f sxd − 2`
x l a
means that for every negative number N there is a positive number such that
if 0 , | x 2 a | , then f sxd , N
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