ASSIGNMENT
112 CHAPTER 1 • Differentiation
If the left-hand and right-hand limits (as x approaches a) are not equal, the limit does not exist at a. If the left-hand and right-hand limits are equal and not infinite, the limit does exist.
A limit xlim f(x) can exist even though the function -Pa value f(a) might not exist. (See Example 3.)
1.1 Exercise Set
For each sequence of numbers, determine the limit and then rewrite the sequence using the notation x a- or x a+. 1. 0.29, 0.299, 0.2999, 0.29999, .. . 2. 1.71, 1.701, 1.7001, 1.70001, … 3. -3.5, -3.05, -3.005, -3.0005, . 4. -4.89, -4.899, -4.8999, -4.89999, 5. 0.6, 0.66, 0.666, 0.6666, … 6. 1.3, 1.33, 1.333, 1.3333, … 7. 0.29, 0.299, 0.2999, 0.29999, … 8. 1.19, 1.199, 1.1999, 1.19999, .. . an. 3 5 9 17 33 25 45 85 18′ 325 • • •
10 3 • 105 1005 10005 10,0005 • • •
Complete each of the following statements.
11. As x approaches , the value of -3x approaches 6.
12. As x approaches , the value of x – 2 approaches 5.
13. The notation is read “the limit, as x approaches 2 from the right.”
14. The notation is read “the limit, as x approaches 3 from the left.”
15. The notation is read “the limit as x approaches 5.”
16. The notation is read “the limit as x approaches I.”
17. The notation x-0lim f(x) is read 4
18. The notation lx-.1 g(x) is read rn
19. The notation Jim F(x) is read x-0 5-
20. The notation x-04 lim G(x) is read
A limitlim f(x) can exist and be different from the x-Pa function value f(a). (See Example 5.)
Graphs and tables are useful tools in determining limits.
For Exercises 21 and 22, consider the function f given by f(x) x – 2, for x s 3, (x – 1, for x > 3. y [.. 4 3 • — 1
-5-4-3-2-1 • { i 1 .R-3 4 • i 1 -1-r-, i!:
If a limit does not exist, state that fact. 11: Find (a)x-, -1 – lim f(x); (b) X lim f(x); (c)
x- 0- lim 1 f(x).
22. Find (a) lim f(x); (b) lim,f(x); (c) lim f(x). For Exercises 23 and 24, consider the function g given by g(x) – fx + 6, for x < – 2, – + 1, for x z -2.
If ct limit does not exist, state that fact. 23. Find (a) x lim4- g(x); (b) lim g(x); (c) lirn g(x). -6 A.-b4 P4 24. Find (a)x-P lim 2- x -• -2 g(x); (b) lim L (c) lim g(x). -2
J
