2.3 Formal Definition of Limit Definition: The statement lim
→
means that for every number |
0, there exists a corresponding number | then|
| 0 such that if 0suchthatforallxintheinterval|
Inexamples1‐3,usethegraphstofinda
guaranteethatthefunctionvalues
willbewithin
0ofthenumberL.
1.
Solution:
Findtheminimumdistancefromc 1toeachendpoint.
25
9
1
16
16
9
7
1
16
16
Therefore,
Thatisif|
1|
7
then √
16
1
|
will
2.
Solution:
Findtheminimumdistancefromc ‐1toeachendpoint.
√3
√3 2
1
0.13397
2
2
2 √5
√5
1
0.1180
2
2
√
Therefore,
Thatisif|
1|
√
then| 4
3|
0.25
3.
Solution:
Findtheminimumdistancefrom
1
1
1
1.99 2 398
1
1
1
2 2.01
402
Therefore,
Thatisif
1
402
then
toeachendpoint.
2
0.01
Inexamples4and5,usetheformaldefinitionofalimittoprovethefollowinglimitstatements.
4.lim 2
→
7
1
Proof:
Givenanumber   0 thenwemustshowthereexistsacorrespondingnumber   0 such
thatwheneverxiswithin  of4then f ( x ) willbewithin  of1.Thatisweneedtoshow
1| whenever|
thereexistsanumber   0 suchthat|
4| .
Scratchwork:
|
1| 2 x  7  1  2 x  8  2 x  4   Choose   
Assume|
2
4|
|
1|
|2
5. lim 3
2
⁄2Then
7
1|
|2
8|
2|
4|
2
→
2
7
Proof:
Givenanumber   0 thenwemustshowthereexistsacorrespondingnumber   0 such
thatwheneverxiswithin  of‐3then f ( x ) willbewithin  of‐7.Thatisweneedtoshow
7 | whenever|
thereexistsanumber   0 suchthat|
3 | .
Scratchwork:
|
7 |
3 x  2   7   3 x  9  3 x  3   Choose   
Assume|
|
3
3|
7 |
|3
3Then
2 7|
|3
9|
3|
3|
3
3