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

Section 2.3 Calculating Limits Using the Limit Laws 101
Other notations for v x b are fxg and :x;.
The greatest integer function is
sometimes called the floor function.
ExamplE 10 The greatest integer function is defined by v x b − the largest integer
that is less than or equal to x. (For instance, v4 b − 4, v4.8b − 4, v b − 3, vs2 b − 1,
v2 1 2b − 21.) Show that lim x l3 v x b does not exist.
y
4
SOLUTION The graph of the greatest integer function is shown in Figure 6. Since
v x b − 3 for 3 < x , 4, we have
3
2
y=[ x]
lim v x b − lim 3 − 3
x l3 1 x l3 1
1
Since v x b − 2 for 2 < x , 3, we have
0
1 2 3
4 5 x
lim v x b − lim 2 − 2
x l3 2 x l3 2
Because these one-sided limits are not equal, lim xl3 v x b does not exist by Theorem 1. ■
FIGURE 6
Greatest integer function
The next two theorems give two additional properties of limits. Their proofs can be
found in Appendix F.
2 Theorem If f sxd < tsxd when x is near a (except possibly at a) and the limits
of f and t both exist as x approaches a, then
lim f sxd < lim tsxd
x l a x l a
3 The Squeeze Theorem If f sxd < tsxd < hsxd when x is near a (except
possibly at a) and
y
L
h
g
then
lim f sxd − lim hsxd − L
x l a x l a
lim tsxd − L
x l a
f
0 a
x
FIGURE 7
The Squeeze Theorem, which is sometimes called the Sandwich Theorem or the
Pinching Theorem, is illustrated by Figure 7. It says that if tsxd is squeezed between
f sxd and hsxd near a, and if f and h have the same limit L at a, then t is forced to have
the same limit L at a.
ExamplE 11 Show that lim
x l 0
x 2 sin 1 x − 0.
SOLUTION First note that we cannot use
lim x 2 sin 1
x l 0 x − lim x 2 ? lim sin 1
x l 0 x l 0 x
because lim x l 0 sins1yxd does not exist (see Example 2.2.4).
Instead we apply the Squeeze Theorem, and so we need to find a function f smaller
than tsxd − x 2 sins1yxd and a function h bigger than t such that both f sxd and hsxd
approach 0. To do this we use our knowledge of the sine function. Because the sine of
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