Exercises: Limits
1–6
Use a table of values to guess the limit.
1. lim x 1/x
2. lim x −
x→∞
3. lim
x→∞
p
x→∞
1 x
1+ √
x
x2 + x
4. lim sin(x2 )
x→∞
x 3 − 2x
5. lim
x→∞
x
6. lim x sin(1/x)
p
x + 25 − 5
7. lim
x
x→0
8. lim
x→∞
x→0
4x − 1
8x − 1
9. Use a table of values to estimate the following limit:
x
x
lim
x→∞ x + 2
15. Let f (x) = x4 .
(a) Find a formula for the average slope of f (x) between x = 2
and x = 2 + h.
(b) Use your answer to part (a) to find the average slope for
h = 0.1, h = 0.01, h = 0.001.
(c) Based on this data, what is the value of f 0 (2)?
√
16. Let f (x) = x.
(a) Find a formula for the average slope of f (x) between x = 1
and x = 1 + h.
(b) Use your answer to part (a) to find the average slope for
h = 0.1, h = 0.01, h = 0.001.
(c) Based on this data, what is the value of f 0 (1)?
17. The following table shows some data points for a function f (x).
Your answer must be correct to four decimal places.
10. Use a table of values to estimate the following limit:
x
lim p
3x2 + 1
x→∞
Your answer must be correct to four decimal places.
11–14 Use numerical or algebraic reasoning to guess the value of
the limit. (Do not use a calculator.)
11. lim
x→∞
13. lim
x→0
1
2x
12. lim
3x + 1
x+4
14. lim
x→∞
x→0
x
f (x)
2
2.1
2.01
2.001
2.0001
3
3.74
3.0524
3.005024
3.00050024
Based on this data, what is the value of f 0 (2)?
Write a limit for the derivative of the given function at the
18–21
given value of x. You do not need to simplify your answer.
x
3x + 1
18. f (x) = x4 , x = 3
19. f (x) =
(x + 5)2 − 25
x
20. f (x) = 2x , x = 5
21. f (x) = 1/x, x = 4
√
3
x, x = 8
Answers
1. 1
11. 0
2. −0.5
12. 1/3
3. ∞
4. no limit
13. 1/4
14. 10
5. −∞
15. (a)
6. 1
7. 0.1
8. 2/3
√
3
8+h − 2
19. lim
h→0
h
10. 0.57735
(2 + h)4 − 16
(b) 34.481, 32.2408, 32.024 (c) 32
h
√
1+h − 1
16. (a)
(b) 0.488088, 0.498756, 0.499875 (c) 0.5
h
(3 + h)4 − 81
18. lim
h→0
h
9. 0.1353
17. 5
25+h − 32
20. lim
h→0
h
1
1
−
4
+
h
4
21. lim
h→0
h