lesson 3: the limit of a function
DESCRIPTION
Limits are where algebra ends and calculus begins.TRANSCRIPT
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Section 1.3The Limit of a Function
V63.0121, Calculus I
January 26–27, 2009
Announcements
I Office Hours: MW 1:30–3:00, TR 1:00–2:00 (WWH 718)
I Blackboard operational
I HW due Wednesday, ALEKS initial due Friday
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Limit
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Zeno’s Paradox
That which is inlocomotion mustarrive at thehalf-way stagebefore it arrives atthe goal.
(Aristotle Physics VI:9,239b10)
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Outline
The Concept of LimitHeuristicsErrors and tolerancesExamplesPathologies
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Heuristic Definition of a Limit
DefinitionWe write
limx→a
f (x) = L
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 toL as we like) by taking x to be sufficiently close to a (on either sideof a) but not equal to a.
![Page 6: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/6.jpg)
The error-tolerance game
A game between two players to decide if a limit limx→a
f (x) exists.
I Player 1: Choose L to be the limit.
I Player 2: Propose an “error” level around L.
I Player 1: Choose a “tolerance” level around a so that x-pointswithin that tolerance level are taken to y -values within theerror level.
If Player 1 can always win, limx→a
f (x) = L.
![Page 7: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/7.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 8: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/8.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 9: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/9.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 10: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/10.jpg)
The error-tolerance game
This tolerance is too big
Still too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 11: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/11.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 12: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/12.jpg)
The error-tolerance game
This tolerance is too big
Still too big
This looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 13: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/13.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 14: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/14.jpg)
The error-tolerance game
This tolerance is too bigStill too big
This looks good
So does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 15: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/15.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks good
So does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 16: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/16.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 17: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/17.jpg)
The error-tolerance game
This tolerance is too bigStill too bigThis looks goodSo does this
a
L
I To be legit, the part of the graph inside the blue (vertical)strip must also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
![Page 18: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/18.jpg)
Example
Find limx→0
x2 if it exists.
SolutionBy setting tolerance equal to the square root of the error, we canguarantee to be within any error.
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Example
Find limx→0
x2 if it exists.
SolutionBy setting tolerance equal to the square root of the error, we canguarantee to be within any error.
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Example
Find limx→0
|x |x
if it exists.
Solution
The function can also be written as
|x |x
=
{1 if x > 0;
−1 if x < 0
What would be the limit?The error-tolerance game fails, but
limx→0+
f (x) = 1 limx→0−
f (x) = −1
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Example
Find limx→0
|x |x
if it exists.
SolutionThe function can also be written as
|x |x
=
{1 if x > 0;
−1 if x < 0
What would be the limit?
The error-tolerance game fails, but
limx→0+
f (x) = 1 limx→0−
f (x) = −1
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The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
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The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 24: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/24.jpg)
The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 25: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/25.jpg)
The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 26: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/26.jpg)
The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 27: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/27.jpg)
The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 28: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/28.jpg)
The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 29: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/29.jpg)
The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 30: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/30.jpg)
The error-tolerance game
x
y
−1
1
Part of graph in-side blue is notinside green
Part of graph in-side blue is notinside green
I These are the only good choices; the limit does not exist.
![Page 31: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/31.jpg)
One-sided limits
DefinitionWe write
limx→a+
f (x) = L
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 toL as we like) by taking x to be sufficiently close to a (on either sideof a) and greater than a.
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One-sided limits
DefinitionWe write
limx→a−
f (x) = L
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 toL as we like) by taking x to be sufficiently close to a (on either sideof a) and less than a.
![Page 33: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/33.jpg)
Example
Find limx→0
|x |x
if it exists.
SolutionThe function can also be written as
|x |x
=
{1 if x > 0;
−1 if x < 0
What would be the limit?The error-tolerance game fails, but
limx→0+
f (x) = 1 limx→0−
f (x) = −1
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Example
Find limx→0+
1
xif it exists.
SolutionThe limit does not exist because the function is unbounded near 0.Next week we will understand the statement that
limx→0+
1
x= +∞
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The error-tolerance game
x
y
0
L?
The graph escapes thegreen, so no good
Even worse!The limit does not existbecause the function isunbounded near 0
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The error-tolerance game
x
y
0
L?
The graph escapes thegreen, so no good
Even worse!The limit does not existbecause the function isunbounded near 0
![Page 37: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/37.jpg)
The error-tolerance game
x
y
0
L?
The graph escapes thegreen, so no good
Even worse!The limit does not existbecause the function isunbounded near 0
![Page 38: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/38.jpg)
The error-tolerance game
x
y
0
L?
The graph escapes thegreen, so no good
Even worse!The limit does not existbecause the function isunbounded near 0
![Page 39: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/39.jpg)
The error-tolerance game
x
y
0
L?
The graph escapes thegreen, so no good
Even worse!The limit does not existbecause the function isunbounded near 0
![Page 40: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/40.jpg)
The error-tolerance game
x
y
0
L?
The graph escapes thegreen, so no good
Even worse!
The limit does not existbecause the function isunbounded near 0
![Page 41: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/41.jpg)
The error-tolerance game
x
y
0
L?
The graph escapes thegreen, so no good
Even worse!
The limit does not existbecause the function isunbounded near 0
![Page 42: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/42.jpg)
Example
Find limx→0+
1
xif it exists.
SolutionThe limit does not exist because the function is unbounded near 0.Next week we will understand the statement that
limx→0+
1
x= +∞
![Page 43: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/43.jpg)
Example
Find limx→0
sin(π
x
)if it exists.
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x
y
−1
1
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What could go wrong?
How could a function fail to have a limit? Some possibilities:
I left- and right- hand limits exist but are not equal
I The function is unbounded near a
I Oscillation with increasingly high frequency near a
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Meet the Mathematician: Augustin Louis Cauchy
I French, 1789–1857
I Royalist and Catholic
I made contributions ingeometry, calculus,complex analysis,number theory
I created the definition oflimit we use today butdidn’t understand it
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Precise Definition of a Limit
Let f be a function defined on an some open interval that containsthe number a, except possibly at a itself. Then we say that thelimit of f (x) as x approaches a is L, and we write
limx→a
f (x) = L,
if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ, then |f (x)− L| < ε.
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The error-tolerance game = ε, δ
L + ε
L− ε
a− δ a + δ
This δ is too big
a− δa + δ
This δ looks good
a− δa + δ
So does this δ
a
L
![Page 49: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/49.jpg)
The error-tolerance game = ε, δ
L + ε
L− ε
a− δ a + δ
This δ is too big
a− δa + δ
This δ looks good
a− δa + δ
So does this δ
a
L
![Page 50: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/50.jpg)
The error-tolerance game = ε, δ
L + ε
L− ε
a− δ a + δ
This δ is too big
a− δa + δ
This δ looks good
a− δa + δ
So does this δ
a
L
![Page 51: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/51.jpg)
The error-tolerance game = ε, δ
L + ε
L− ε
a− δ a + δ
This δ is too big
a− δa + δ
This δ looks good
a− δa + δ
So does this δ
a
L
![Page 52: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/52.jpg)
The error-tolerance game = ε, δ
L + ε
L− ε
a− δ a + δ
This δ is too big
a− δa + δ
This δ looks good
a− δa + δ
So does this δ
a
L
![Page 53: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/53.jpg)
The error-tolerance game = ε, δ
L + ε
L− ε
a− δ a + δ
This δ is too big
a− δa + δ
This δ looks good
a− δa + δ
So does this δ
a
L
![Page 54: Lesson 3: The Limit of a Function](https://reader033.vdocument.in/reader033/viewer/2022052505/5562d896d8b42aac778b4bf1/html5/thumbnails/54.jpg)
The error-tolerance game = ε, δ
L + ε
L− ε
a− δ a + δ
This δ is too big
a− δa + δ
This δ looks good
a− δa + δ
So does this δ
a
L