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

84 Chapter 2 Limits and Derivatives
Notice the phrase “but x ± a” in the definition of limit. This means that in find ing the
limit of f sxd as x approaches a, we never consider x − a. In fact, f sxd need not even be
defined when x − a. The only thing that matters is how f is defined near a.
Figure 2 shows the graphs of three functions. Note that in part (c), f sad is not defined
and in part (b), f sad ± L. But in each case, regardless of what happens at a, it is true
that lim x l a f sxd − L.
y
y
y
L
L
L
0
a
x
0
a
x
0
a
x
(a)
(b)
(c)
figure 2 lim f sxd − L in all three cases
x l a
x , 1
f sxd
0.5 0.666667
0.9 0.526316
0.99 0.502513
0.999 0.500250
0.9999 0.500025
x . 1 f sxd
1.5 0.400000
1.1 0.476190
1.01 0.497512
1.001 0.499750
1.0001 0.499975
ExamplE 1 Guess the value of lim
x l1
x 2 1
x 2 2 1 .
SOLUTION Notice that the function f sxd − sx 2 1dysx 2 2 1d is not defined when x − 1,
but that doesn’t matter because the definition of lim x l a f sxd says that we consider
values of x that are close to a but not equal to a.
The tables at the left give values of f sxd (correct to six decimal places) for values of
x that approach 1 (but are not equal to 1). On the basis of the values in the tables, we
make the guess that
lim
x l 1
x 2 1
x 2 2 1 − 0.5
Example 1 is illustrated by the graph of f in Figure 3. Now let’s change f slightly by
giving it the value 2 when x − 1 and calling the resulting function t:
tsxd −Hx 2 1
x 2 2 1
if x ± 1
2 if x − 1
■
1 0.5
This new function t still has the same limit as x approaches 1. (See Figure 4.)
y
y
2
x-1
y=
≈-1
y=©
0.5
0.5
0 1
x
0 1
x
figure 3 Figure 4
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