Worksheet: 01.02
Topic: Introduction to Limits
Due: Sept 8
Answer the following conceptual questions
1. (Fill in the blank) Let s(t) be a position function of some object. Then the average velocity over some time
of s(t).
interval can be graphically interpreted as a slope of the
2. Describe a real world situation using a limit.
3. If asked to estimate the limit lim f (x), describe the strategy you would use estimate it.
x→c
4. Explain the difference between a right-hand limit and a left-hand limit.
5. (T/F) Velocity can be geometrically interpreted as a tangent line.
Approximate the given limits numerically
6. lim
x→0
x+1
x2 + 3x
x2 − 2x − 3
x→3 x2 − 4x + 3
7. lim
A function f and a value a are given. Approximate lim
h→0
8. f (x) = −7x + 2,
f (a + h) − f (a)
, using h = ±0.1, ±0.01.
h
a=3
9. f (x) = 9x + 0.06, a = −1
10. f (x) =
1
, a=2
x+1
11. f (x) = ln x, a = 5
For problems 12-19, use the graph of f (x) to compute the following:
12.
lim
x→−1+
f (x)
13. lim− f (x)
x→3
14. lim− f (x)
x→2
15. lim+ f (x)
x→0
16. lim− f (x)
x→1
17. lim+ f (x)
x→2
18. f (2)
19. f (1)
1
For problems 20-27, use the graph of g(x) to compute the following:
20.
lim
x→−1+
g(x)
21. lim+ g(x)
x→1
22. lim+ g(x)
x→2
23. lim− g(x)
x→1
24. lim− g(x)
x→2
25. lim+ g(x)
x→0
26. g(1)
27. g(2)
Answer the following conceptual questions
1. Not given, read the notes
For problems 20-27, use the graph of g(x) to compute
the following:
3. Not given, read the notes
21. 1
5. Not given, read the notes
23. 2
Approximate the given limits numerically
7. 2
25. 0
A function f and a value a are given. Approximate
f (a + h) − f (a)
lim
, using h = ±0.1, ±0.01.
h→0
h
f (a+h)−f (a)
h
h
−0.1
9
9. −0.01
9
0.01
9
0.1
9
The limit seems to be exactly 9.
f (a+h)−f (a)
h
h
−0.1
11. −0.01
0.01
0.1
The limit is
0.202027
0.2002
0.1998
0.198026
approx. 0.2.
For problems 12-19, use the graph of f (x) to compute
the following:
13. 0
15. 0
17. 1
19. 0
27. 2