The limit is where algebra ends and caclulus begins. We describe the definition of limit as a game to find the acceptable tolerance for each error.

1. Section 1.1–1.3
Functions and the concept of limit
V63.0121, Calculus I
September 9, 2009
Announcements
Syllabus is on the common Blackboard
Office Hours TBA
Read Sections 1.1–1.3 of the textbook this week.
. . . . . .

2. Outline
Functions
Functions expressed by formulas
Functions expressed by data
Functions described graphically
Functions described verbally
Classes of Functions
Limits
Heuristics
Errors and tolerances
Examples
Pathologies
. . . . . .

3. What is a function?
Definition
A function f is a relation which assigns to to every element x in a
set D a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { f(x) | x ∈ D } is called the range of f.
. . . . . .

4. The Modeling Process
. .
Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
. . . . . .

5. Plato’s Cave
. . . . . .

6. Functions expressed by formulas
Any expression in a single variable x defines a function. In this
case, the domain is understood to be the largest set of x which
after substitution, give a real number.
. . . . . .

7. Functions expressed by data
In science, functions are often defined by data. Or, we observe
data and assume that it’s close to some nice continuous function.
. . . . . .

8. Example
Here is the temperature in Boise, Idaho measured in 15-minute
intervals over the period August 22–29, 2008.
.
1
. 00 .
9
.0.
8
.0.
7
.0.
6
.0.
5
.0.
4
.0.
3
.0.
2
.0.
1
.0. . . . . . . .
8
. /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29
8 8 8 8 8 8 8
. . . . . .

9. Functions described graphically
Sometimes all we have is the “picture” of a function, by which
we mean, its graph.
.
.
. . . . . .

10. Functions described graphically
Sometimes all we have is the “picture” of a function, by which
we mean, its graph.
.
.
The one on the right is a relation but not a function.
. . . . . .

11. Functions described verbally
Oftentimes our functions come out of nature and have verbal
descriptions:
The temperature T(t) in this room at time t.
The elevation h(θ) of the point on the equation at longitude
θ.
The utility u(x) I derive by consuming x burritos.
. . . . . .

12. Classes of Functions
linear functions, defined by slope an intercept, point and
point, or point and slope.
quadratic functions, cubic functions, power functions,
polynomials
rational functions
trigonometric functions
exponential/logarithmic functions
. . . . . .

13. Outline
Functions
Functions expressed by formulas
Functions expressed by data
Functions described graphically
Functions described verbally
Classes of Functions
Limits
Heuristics
Errors and tolerances
Examples
Pathologies
. . . . . .

14. Limit
. . . . . .

15. Zeno’s Paradox
That which is in
locomotion must
arrive at the
half-way stage
before it arrives at
the goal.
(Aristotle Physics VI:9,
239b10)
. . . . . .

16. Heuristic Definition of a Limit
Definition
We write
lim f(x) = L
x→a
and say
“the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) but not equal to a.
. . . . . .

17. The error-tolerance game
A game between two players to decide if a limit lim f(x) exists.
x→a
Player 1: Choose L to be the limit.
Player 2: Propose an “error” level around L.
Player 1: Choose a “tolerance” level around a so that
x-points within that tolerance level are taken to y-values
within the error level.
If Player 1 can always win, lim f(x) = L.
x→a
. . . . . .

18. The error-tolerance game
L
.
.
a
.
. . . . . .

19. The error-tolerance game
L
.
.
a
.
. . . . . .

20. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .

21. The error-tolerance game
T
. his tolerance is too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .

22. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .

23. The error-tolerance game
S
. till too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .

24. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .

25. The error-tolerance game
T
. his looks good
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .

26. The error-tolerance game
S
. o does this
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .

27. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
If Player 2 shrinks the error, Player 1 can still win.
. . . . . .

28. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
If Player 2 shrinks the error, Player 1 can still win.
. . . . . .

29. Example
Find lim x2 if it exists.
x→0
. . . . . .

30. Example
Find lim x2 if it exists.
x→0
Solution
By setting tolerance equal to the square root of the error, we can
guarantee to be within any error.
. . . . . .

31. Example
|x|
Find lim if it exists.
x→0 x
. . . . . .

32. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
. . . . . .

33. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

34. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

35. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

36. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

37. The error-tolerance game
y
.
. .
1
. x
.
.
Part of graph in-
. 1.
− side blue is not
inside green
. . . . . .

38. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

39. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

40. The error-tolerance game
y
.
.
Part of graph in-
side blue is not . .
1
inside green
. x
.
. 1.
−
. . . . . .

41. The error-tolerance game
y
.
.
Part of graph in-
side blue is not . .
1
inside green
. x
.
. 1.
−
These are the only good choices; the limit does not exist.
. . . . . .

42. One-sided limits
Definition
We write
lim f(x) = L
x→a+
and say
“the limit of f(x), as x approaches a from the right, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) and greater than a.
. . . . . .

43. One-sided limits
Definition
We write
lim f(x) = L
x→a−
and say
“the limit of f(x), as x approaches a from the left, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) and less than a.
. . . . . .

44. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
The error-tolerance game fails, but
lim f(x) = 1 lim f(x) = −1
x→0+ x→0−
. . . . . .

45. Example
1
Find lim if it exists.
x→0+ x
. . . . . .

46. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

47. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

48. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

49. The error-tolerance game
y
.
.
The graph escapes the
green, so no good
.? .
L
. x
.
0
.
. . . . . .

50. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

51. The error-tolerance game
y
.
E
. ven worse!
.? .
L
. x
.
0
.
. . . . . .

52. The error-tolerance game
y
.
.
The limit does not exist
because the function is
unbounded near 0
.? .
L
. x
.
0
.
. . . . . .

53. Example
1
Find lim if it exists.
x→0+ x
Solution
The limit does not exist because the function is unbounded near
0. Next week we will understand the statement that
1
lim = +∞
x→0+ x
. . . . . .

54. Example (π )
Find lim sin if it exists.
x→0 x
. . . . . .

55. Example (π )
Find lim sin if it exists.
x→0 x
y
.
. .
1
. x
.
. 1.
−
. . . . . .

56. What could go wrong?
How could a function fail to have a limit? Some possibilities:
left- and right- hand limits exist but are not equal
The function is unbounded near a
Oscillation with increasingly high frequency near a
. . . . . .

57. Meet the Mathematician: Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contributions in
geometry, calculus,
complex analysis,
number theory
created the definition of
limit we use today but
didn’t understand it
. . . . . .

58. Precise Definition of a Limit
No, this is not going to be on the test Let f be a function defined
on an some open interval that contains the number a, except
possibly at a itself. Then we say that the limit of f(x) as x
approaches a is L, and we write
lim f(x) = L,
x→a
if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ , then |f(x) − L| < ε.
. . . . . .

59. The error-tolerance game = ε, δ
L
.
.
a
.
. . . . . .

60. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
a
.
. . . . . .

61. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
aa
. − δ. . + δ
a
. . . . . .

62. The error-tolerance game = ε, δ
T
. his δ is too big
L
. +ε
L
.
. −ε
L
.
aa
. − δ. . + δ
a
. . . . . .

63. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
a . a+
. −. δ δ
a
. . . . . .

64. The error-tolerance game = ε, δ
T
. his δ looks good
L
. +ε
L
.
. −ε
L
.
a . a+
. −. δ δ
a
. . . . . .

65. The error-tolerance game = ε, δ
S
. o does this δ
L
. +ε
L
.
. −ε
L
.
a
aa .+
. .− δ δ
. . . . . .