Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.

1. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Sec on 1.3 Notes
The Limit of a Func on
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
January 31, 2011
Announcements
First wri en HW due Wednesday February 2
.
. Get-to-know-you survey and photo deadline is February 11 .
Announcements Notes
First wri en HW due
Wednesday February 2
Get-to-know-you survey
and photo deadline is
February 11
. .
Guidelines for written homework Notes
Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (001), recita on number,
date, assignment number, book sec ons.
Explain your work and your reasoning in your own words. Use
complete English sentences.
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2. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Rubric Notes
Points Descrip on of Work
3 Work is completely accurate and essen ally perfect.
Work is thoroughly developed, neat, and easy to read.
Complete sentences are used.
2 Work is good, but incompletely developed, hard to
read, unexplained, or jumbled. Answers which are
not explained, even if correct, will generally receive 2
points. Work contains “right idea” but is flawed.
1 Work is sketchy. There is some correct work, but most
of work is incorrect.
0 Work minimal or non-existent. Solu on is completely
incorrect.
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Written homework: Don’t Notes
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Written homework: Do Notes
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3. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Written homework: Do Notes
Written Explanations
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Written homework: Do Notes
Graphs
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Objectives Notes
Understand and state the
informal defini on of a
limit.
Observe limits on a
graph.
Guess limits by algebraic
manipula on.
Guess limits by numerical
informa on.
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4. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Notes
Limit
.
. .
Zeno’s Paradox Notes
That which is in locomo on must
arrive at the half-way stage before
it arrives at the goal.
(Aristotle Physics VI:9, 239b10)
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Outline Notes
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
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5. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Heuristic Definition of a Limit Notes
Defini on
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.
. .
Outline Notes
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
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The error-tolerance game Notes
A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
Step 1 Dana proposes L to be the limit.
Step 2 Emerson challenges with an “error” level around L.
Step 3 Dana chooses a “tolerance” level around a so that points x
within that tolerance of a (not coun ng a itself) are taken to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.
Step 4 If Dana’s move is a good one, Emerson can challenge again
or give up. If Emerson gives up, Dana wins and the limit is L.
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6. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
The error-tolerance game Notes
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can s ll move.
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Outline Notes
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
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Playing the E-T Game Notes
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs
to guarantee that −0.01 < x2 < 0.01 for all x sufficiently close
to zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
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7. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Playing the E-T Game Notes
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?
A tolerance of 0.01 works because
|x| < 10−2 =⇒ x2 < 10−4 .
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Playing the E-T Game Notes
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana has a shortcut: By se ng tolerance equal to the square
root of the error, Dana can win every round. Once Emerson
realizes this, Emerson must give up.
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Graphical version of E-T game Notes
with x2
y
No ma er how small an
error Emerson picks,
Dana can find a fi ng
tolerance band.
.
x
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8. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
A piecewise-defined function Notes
Example
|x|
Find lim if it exists.
x→0 x
Solu on
The func on can also be wri en as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
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The E-T game with a piecewise Notes
function
|x|
Find lim if it exists.
x→0 x y
1
. x
−1
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One-sided limits Notes
Defini on
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.
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9. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
One-sided limits Notes
Defini on
We write
lim f(x) = L
x→a+−
and say
“the limit of f(x), as x approaches a from the le , 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.
. .
Another Example Notes
Example
1
Find lim+ if it exists.
x→0 x
Solu on
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The error-tolerance game with 1/x Notes
y
1
Find lim+ if it exists. L?
x→0 x
. x
0
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10. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Weird, wild stuff Notes
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
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Function values Notes
x π/x sin(π/x)
1 π π/2
1/2 2π
1/k kπ
2 π/2
2/5 5π/2 π . 0
2/9 9π/2
2/13 13π/2
2/3 3π/2
2/7 7π/2 3π/2
2/11 11π/2
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What could go wrong? Notes
Summary of Limit Pathologies
How could a func on fail to have a limit? Some possibili es:
le - and right- hand limits exist but are not equal
The func on is unbounded near a
Oscilla on with increasingly high frequency near a
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11. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
Meet the Mathematician Notes
Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contribu ons in geometry,
calculus, complex analysis,
number theory
created the defini on of limit
we use today but didn’t
understand it
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Outline Notes
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
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Precise Definition of a Limit Notes
Let f be a func on 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.
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12. . V63.0121.001: Calculus I
. Sec on 1.3:. Limits January 31, 2011
The error-tolerance game = ε, δ Notes
L+ε
L
L−ε
.
a−δ a a+δ
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Summary Notes
Many perspectives on limits
Graphical: L is the value the func on
“wants to go to” near a
Heuris cal: f(x) can be made arbitrarily
close to L by taking x sufficiently close
to a.
Informal: the error/tolerance game
Precise: if for every ε > 0 there is a
corresponding δ > 0 such that if
0 < |x − a| < δ, then |f(x) − L| < ε.
Algebraic: next me
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Notes
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