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

88 Chapter 2 Limits and Derivatives
2 Definition of One-Sided Limits We write
lim f sxd − L
x la 2
and say the left-hand limit of f sxd as x approaches a [or the limit of f sxd as
x approaches a from the left] is equal to L if we can make the values of f sxd
arbitrarily close to L by taking x to be sufficiently close to a with x less than a.
Notice that Definition 2 differs from Definition 1 only in that we require x to be less
than a. Similarly, if we require that x be greater than a, we get “the right-hand limit of
f sxd as x approaches a is equal to L” and we write
lim
x la 1
f sxd − L
Thus the notation x l a 1 means that we consider only x greater than a. These definitions
are illustrated in Figure 9.
y
y
ƒ
L
L
ƒ
FIGURE 9
0 x a x
(a) lim ƒ=L
x a _
0 a x
x
(b) lim ƒ=L
x a +
By comparing Definition l with the definitions of one-sided limits, we see that the
following is true.
3 lim
x l a
f sxd − L if and only if lim
x la 2 f sxd − L and
lim
x la 1
f sxd − L
y
4
3
y=©
1
0
1 2 3 4 5 x
FIGURE 10
ExamplE 7 The graph of a function t is shown in Figure 10. Use it to state the values
(if they exist) of the following:
(a) lim tsxd
x l 2 2
(d) lim tsxd
x l 5 2
(b) lim tsxd
x l 2 1
(e) lim tsxd
x l 5 1
(c) lim tsxd
x l 2
(f) lim tsxd
x l 5
SOLUTION From the graph we see that the values of tsxd approach 3 as x approaches 2
from the left, but they approach 1 as x approaches 2 from the right. Therefore
(a) lim tsxd − 3 and (b) lim tsxd − 1
x l 22 x l 2 1
(c) Since the left and right limits are different, we conclude from (3) that lim x l 2 tsxd
does not exist.
The graph also shows that
(d) lim tsxd − 2 and (e) lim tsxd − 2
x l 52 x l 5 1
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