Thomas Calculus 13th [Solutions]

Section 2.3 The Precise Definition of a Limit 77
36. lim 4 4
2
x 2
2
x
Step 1: 4 2 0.4 0.4 4 2 0.4 1.6 4 2.4 10 x 10 10 10
x
x
x 16 4 24 4 x 6
or 5 5
3 x 2 .
Step 2: x 2 x 2
2 x 2.
Then 2 5 1 , or 2 5 1
3 3
2 2 ; thus 1
3 .
37. Step 1: (9 x) 5 4 x 4 x 4 4 x 4 4 x 4 .
Step 2: x 4 x 4
4 x 4.
Then 4 4 , or 4 4 . Thus choose .
38. Step 1: (3x
7) 2 3x
9 9 3x
9 3 x 3 .
3 3
Step 2: x 3 x 3
3 x 3.
Then 3 3 , or 3 3 . Thus choose
3 3
3 3
3 .
39. Step 1: x 5 2 x 5 2 2 x 5 2
2 2
(2 ) 5 x (2 ) 5.
Step 2: x 9 x 9
9 x 9.
Then
distance,
2 2
9 4 9 4 , or
2
4 .
40. Step 1: 4 x 2 4 x 2 2 4 x 2
2 2
(2 ) x 5 (2 )
2 2
9 4 9 4 . Thus choose the smaller
2 2
(2 ) 4 x (2 )
2 2 2 2
(2 ) x 4 (2 ) (2 ) 4 x (2 ) 4.
Step 2: x 0 x .
2 2
2
2 2
Then (2 ) 4 4 4 , or (2 ) 4 4 . Thus choose the
2
smaller distance, 4 .
41. Step 1: For
2 2
x 1, x 1 x 1
2
1 x 1 1 x 1
1 x 1 near x 1.
Step 2: x 1 x 1 1 x 1.
Then 1 1 1 1 , or 1 1 1 1. Choose
min 1 1 , 1 1 , that is, the smaller of the two distances.
42. Step 1: For
2 2
x 2, x 4 x 4
2
4 x 4 4 x 4 4 x
4 near x 2.
Step 2: x ( 2) x 2
2 x 2.
Then 2 4 4 2, or 2 4 2 4 . Choose
min 4 2, 2 4 .
43. Step 1:
1
1
1
1 1
1
1
1
x
1
.
x
x
x
1 1
Step 2: x 1 x 1 1 x 1 .
Then 1
1
1
1
, or 1
1 1
1 .
1 1 1
1 1 1
Choose , the smaller of the two distances.
1
1 1 1 1 1
1 1 1 3 1 1 3
44. Step 1:
2 3 2
x
x 3 3 2 3 3 2
x
x 3
3 x 2 3 3 x 3
or 3
x
3
for x near 3.
1 3 1 3 1 3 1 3 ,
1 3 1 3
Copyright
2014 Pearson Education, Inc.