Review Exercises for Chapter 1
Review Exercises for Chapter 1 1. Calculus required. Using a graphing utility, you can estimate the length to be 8.3. Or, the length is slightly longer than the distance between the two points, 8.25. 3.
x
0.1
0.01 0.001
f x
0.26
0.25
0.250
1
0.001
0.01
0.1
0.2499
0.249
0.24
−1
lim f x 0.25
x→0
5. h x
1
−1
x2 2x x
7. lim 3 x 3 1 2
(a) lim h x 2
x→1
x→0
Let > 0 be given. Choose . Then for
(b) lim h x 3
0 < x 1 < , you have
x→ 1
x 1 < 1 x < 3 x 2 < f x L < 9. lim x2 3 1 x→2
1
x 2 .
Let > 0 be given. We need x2 3 1 < ⇒ x2 4 x 2 x 2 < ⇒ x 2 < Assuming, 1 < x < 3, you can choose 5. Hence, for 0 < x 2 < 5 you have
1
x 2 < 5 < x 2
x 2 x 2 < x2 4 < x2 3 1 < f x L <
11. lim t 2 4 2 6 2.45
13. lim
t→ 2
t→4
15. lim
x→4
x 2
x 4
lim
x→4
lim
x→4
x 2
17. lim
x 2 x 2 1 x 2
1 4 2
x→0
1 4
x3 125 x 5 x2 5x 25 lim x→ 5 x 5 x→ 5 x 5
19. lim
lim x2 5x 25 x→ 5
75
21. lim
x→0
1 t 2 1 lim t2 4 t→ 2 t 2 4
1 x 1 1 1 x 1 lim x→0 x x x 1 1 1 lim x→0 x 1
1 cos x x lim x→0 sin x sin x
1 xcos x 1 0 0
47