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

102 Chapter 2 Limits and Derivatives
any number lies between 21 and 1, we can write.
4 21 < sin 1 x < 1
Any inequality remains true when multiplied by a positive number. We know that
x 2 > 0 for all x and so, multiplying each side of the inequalities in (4) by x 2 , we get
y
y=≈
as illustrated by Figure 8. We know that
2x 2 < x 2 sin 1 x < x 2
FIGURE 8
y − x 2 sins1yxd
0 x
y=_≈
lim x 2 − 0 and lim s2x 2 d − 0
x l 0 x l 0
Taking f sxd − 2x 2 , tsxd − x 2 sins1yxd, and hsxd − x 2 in the Squeeze Theorem, we
obtain
lim x 2 sin 1
x l 0 x − 0
■
1. Given that
lim f sxd − 4
x l 2
lim tsxd − 22 lim
x l 2
hsxd − 0
x l 2
find the limits that exist. If the limit does not exist, explain why.
(a) lim f f sxd 1 5tsxdg
x l 2
(c) lim sf sxd
x l 2
tsxd
(e) lim
x l2 hsxd
(b) lim
x l 2 ftsxdg3
3f sxd
(d) lim
x l 2 tsxd
tsxdhsxd
(f) lim
x l 2 f sxd
2. The graphs of f and t are given. Use them to evaluate each
limit, if it exists. If the limit does not exist, explain why.
(a) lim
x l2
f f sxd 1 tsxdg
(c) lim
x l21 f f sxdtsxdg
(e) lim
x l2
fx 2 f sxdg
y
1
0 1
y=ƒ
x
(b) lim
x l 0
f f sxd 2 tsxdg
(d) lim
x l3
f sxd
tsxd
(f) f s21d 1 lim
x l21 tsxd
0 1
y=©
3–9 Evaluate the limit and justify each step by indicating the
appropriate Limit Law(s).
3. lim
x l3
s5x 3 2 3x 2 1 x 2 6d
y
1
x
4. lim
xl 21 sx 4 2 3xdsx 2 1 5x 1 3d
t 4 2 2
5. lim
t l 22 2t 2 2 3t 1 2
7. lim s1 1 s 3 x ds2 2 6x 2 1 x 3 d
x l 8
9. lim
x l 2Î 2x 2 1 1
3x 2 2
10. (a) What is wrong with the following equation?
x 2 1 x 2 6
x 2 2
− x 1 3
6. lim su 4 1 3u 1 6
ul22
8. lim
t l 2S t 2 2
2 2
t 3 2 3t 1 5D
(b) In view of part (a), explain why the equation
is correct.
x 2 1 x 2 6
lim
x l2 x 2 2
11–32 Evaluate the limit, if it exists.
x 2 2 6x 1 5
11. lim
x l5 x 2 5
x 2 2 5x 1 6
13. lim
x l5 x 2 5
15.
lim
t l23
t 2 2 9
2t 2 1 7t 1 3
17. lim
h l 0
s25 1 hd 2 2 25
h
− lim
x l2
sx 1 3d
x 2 1 3x
12. lim
x l23 x 2 2 x 2 12
14. lim
x l 4
x 2 1 3x
x 2 2 x 2 12
2x 2 1 3x 1 1
16. lim
x l21 x 2 2 2x 2 3
s2 1 hd 3 2 8
18. lim
h l 0 h
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