1.
Section 1.3
The Concept of Limit
V63.0121.002.2010Su, Calculus I
New York University
May 18, 2010
Announcements
WebAssign Class Key: nyu 0127 7953
Office Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here)
Quiz 1 Thursday on 1.1–1.4
. . . . . .
2.
Announcements
WebAssign Class Key: nyu
0127 7953
Office Hours: MR
5:00–5:45, TW 7:50–8:30,
CIWW 102 (here)
Quiz 1 Thursday on
1.1–1.4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 2 / 32
3.
Objectives
Understand and state the
informal definition of a limit.
Observe limits on a graph.
Guess limits by algebraic
manipulation.
Guess limits by numerical
information.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 3 / 32
4.
Last Time
Key concept: function
Properties of functions: domain and range
Kinds of functions: linear, polynomial, power, rational, algebraic,
transcendental.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 4 / 32
6.
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)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 5 / 32
7.
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 6 / 32
8.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 7 / 32
9.
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 8 / 32
10.
The error-tolerance game
A game between two players to decide if a limit lim f(x) exists.
x→a
Step 1 Player 1 proposes L to be the limit.
Step 2 Player 2 chooses an “error” level around L: the maximum
amount f(x) can be away from L.
Step 3 Player 1 looks for a “tolerance” level around a: the maximum
amount x can be from a while ensuring f(x) is within the given
error of L. The idea is that points x within the tolerance level of
a are taken by f to y-values within the error level of L, with the
possible exception of a itself.
If Player 1 can do this, he wins the round. If he cannot, he
loses the game: the limit cannot be L.
Step 4 Player 2 go back to Step 2 with a smaller error. Or, he can give
up and concede that the limit is L.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 9 / 32
11.
The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
12.
The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
13.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
14.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
15.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
16.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
17.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
18.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
19.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
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.
If Player 2 shrinks the error, Player 1 can still win.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
21.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
22.
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 11 / 32
23.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
24.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
Solution
Step 1 Player 1: I claim the limit is zero.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
25.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
Solution
Step 1 Player 1: I claim the limit is zero.
Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
26.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
Solution
Step 1 Player 1: I claim the limit is zero.
Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.
Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
27.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
Solution
Step 1 Player 1: I claim the limit is zero.
Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.
Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.
Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
28.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
Solution
Step 1 Player 1: I claim the limit is zero.
Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.
Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.
Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?
Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
29.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
Solution
Step 1 Player 1: I claim the limit is zero.
Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.
Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.
Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?
Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
30.
Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0
Solution
Step 1 Player 1: I claim the limit is zero.
Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.
Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.
Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?
Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, by
setting the tolerance equal to the square root of the error, Player 1 can
always win. Player 2 should give up and concede that the limit is 0. . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
31.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
32.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
33.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
34.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
35.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
36.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
37.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
38.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
39.
Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
No matter how small an error band Player 2 picks, Player 1 can
find a fitting tolerance band.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
40.
Limit of a piecewise function
Example
|x|
Find lim if it exists.
x→0 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
41.
Limit of a piecewise function
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?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
42.
The E-T game with a piecewise function
y
.
.
. . .
1
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
43.
The E-T game with a piecewise function
y
.
.
. . .
1
I
. think the limit is 1
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
44.
The E-T game with a piecewise function
y
.
.
. . .
1
I
. think the limit is 1
. . ..
x
C
. an you fit an error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
45.
The E-T game with a piecewise function
y
.
.
. . .
1
H
. ow about this for a tolerance?
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
46.
The E-T game with a piecewise function
y
.
.
. . .
1
H
. ow about this for a tolerance?
. . ..
x
.
No. Part of
graph inside
. 1.
−
blue is not inside
green
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
47.
The E-T game with a piecewise function
y
.
.
. . .
1
O
. h, I guess the limit isn’t 1
. . ..
x
.
No. Part of
graph inside
. 1.
−
blue is not inside
green
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
48.
The E-T game with a piecewise function
y
.
.
. . .
1
. think the limit is −1
I
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
49.
The E-T game with a piecewise function
y
.
.
. . .
1
. think the limit is −1
I
. . ..
x
C
. an you fit an error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
50.
The E-T game with a piecewise function
y
.
.
. . .
1
H
. ow about this for a tolerance?
. . ..
x
C
. an you fit an error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
51.
The E-T game with a piecewise function
y
.
.
.
No. Part of
. graph inside
. .
1
blue is not inside
. ow about this for a tolerance? green
H
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
52.
The E-T game with a piecewise function
y
.
.
.
No. Part of
. graph inside
. .
1
blue is not inside
. h, I guess the limit isn’t −1
O green
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
53.
The E-T game with a piecewise function
y
.
.
. . .
1
I
. think the limit is 0
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
54.
The E-T game with a piecewise function
y
.
.
. . .
1
I
. think the limit is 0
. . ..
x
C
. an you fit an error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
55.
The E-T game with a piecewise function
y
.
.
. . .
1
H
. ow about this for a tolerance?
. . ..
x
C
. an you fit an error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
56.
The E-T game with a piecewise function
y
.
.
. . .
1
H
. ow about this for a tolerance?
. . ..
x
.
No. None of
. 1.
−
graph inside blue
is inside green
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
57.
The E-T game with a piecewise function
y
.
.
. . .
1
.
. Oh, I guess the . ..
x
limit isn’t 0
.
No. None of
. 1.
−
graph inside blue
is inside green
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
58.
The E-T game with a piecewise function
y
.
.
. . .
1
.
I give up! I
. guess there’s . ..
x
no limit!
. 1.
−
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
59.
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 and greater than a.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
60.
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 and less than a.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
61.
The error-tolerance game on the right
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
62.
The error-tolerance game on the right
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
63.
The error-tolerance game on the right
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
64.
The error-tolerance game on the right
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
65.
The error-tolerance game on the right
y
.
. .
1
. x
.
.
All of graph in-
. 1.
− side blue is in-
side green
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
66.
The error-tolerance game on the right
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
67.
The error-tolerance game on the right
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
68.
The error-tolerance game on the right
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
69.
The error-tolerance game on the right
y
.
.
All of graph in- . .
1
side blue is in-
side green
. x
.
. 1.
−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
70.
The error-tolerance game on the right
y
.
.
All of graph in- . .
1
side blue is in-
side green
. x
.
. 1.
−
So lim+ f(x) = 1 and lim f(x) = −1
x→0 x→0−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
71.
Limit of a piecewise function
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−
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 18 / 32
72.
Another Example
Example
1
Find lim+ if it exists.
x→0 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 19 / 32
73.
The error-tolerance game with lim (1/x)
x→0
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
74.
The error-tolerance game with lim (1/x)
x→0
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
75.
The error-tolerance game with lim (1/x)
x→0
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
76.
The error-tolerance game with lim (1/x)
x→0
y
.
.
The graph escapes
the green, so no good
.?.
L
. x
.
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
77.
The error-tolerance game with lim (1/x)
x→0
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
78.
The error-tolerance game with lim (1/x)
x→0
y
.
E
. ven worse!
.?.
L
. x
.
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
79.
The error-tolerance game with lim (1/x)
x→0
y
.
.
The limit does not ex-
ist because the func-
tion is unbounded near
0
.?.
L
. x
.
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
80.
Another (Bad) Example: Unboundedness
Example
1
Find lim+ if it exists.
x→0 x
Solution
The limit does not exist because the function is unbounded near 0.
Later we will talk about the statement that
1
lim+ = +∞
x→0 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 21 / 32
81.
Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 22 / 32
83.
Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
84.
Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
f(x) = 0 when x =
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
85.
Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
86.
Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
f(x) = −1 when x =
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
87.
Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
2
f(x) = −1 when x = for any integer k
4k − 1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
88.
Weird, wild stuff continued
Here is a graph of the function:
y
.
. .
1
. x
.
. 1.
−
There are infinitely many points arbitrarily close to zero where f(x) is 0,
or 1, or −1. So the limit cannot exist.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 25 / 32
89.
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 26 / 32
90.
What could go wrong?
Summary of Limit Pathologies
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 27 / 32
91.
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 28 / 32
92.
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 29 / 32
93.
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 30 / 32
94.
The error-tolerance game = ε, δ
L
.
.
a
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
95.
The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
a
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
96.
The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
97.
The error-tolerance game = ε, δ
T
. his δ is too big
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
98.
The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
99.
The error-tolerance game = ε, δ
T
. his δ looks good
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
100.
The error-tolerance game = ε, δ
S
. o does this δ
L
. +ε
L
.
. −ε
L
.
. .− δ δ
aa .+
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
101.
Summary
y
.
Fundamental Concept: . .
1
limit
Error-Tolerance game . x
.
gives a methods of arguing
limits do or do not exist
Limit FAIL: jumps,
. 1.
−
unboundedness, sin(π/x)
FAIL
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 32 / 32
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