lesson 3: the limit of a function (handout)

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Sec on 1.3The Limit of a Func on

V63.0121.001: Calculus IProfessor Ma hew Leingang

New York University

January 31, 2011

Announcements

I First wri en HW due Wednesday February 2I Get-to-know-you survey and photo deadline is February 11

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Announcements

I First wri en HW dueWednesday February 2

I Get-to-know-you surveyand photo deadline isFebruary 11

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Guidelines for written homework

I Papers should be neat and legible. (Use scratch paper.)I Label with name, lecture number (001), recita on number,date, assignment number, book sec ons.

I Explain your work and your reasoning in your own words. Usecomplete English sentences.

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. Sec on 1.3: Limits. V63.0121.001: Calculus I . January 31, 2011

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RubricPoints Descrip on of Work3 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 toread, unexplained, or jumbled. Answers which arenot explained, even if correct, will generally receive 2points. Work contains “right idea” but is flawed.

1 Work is sketchy. There is some correct work, but mostof work is incorrect.

0 Work minimal or non-existent. Solu on is completelyincorrect.

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Written homework: Don’t

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Written homework: Do

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Written homework: DoWritten Explanations

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Written homework: DoGraphs

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ObjectivesI Understand and state theinformal defini on of alimit.

I Observe limits on agraph.

I Guess limits by algebraicmanipula on.

I Guess limits by numericalinforma on.

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Limit

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Zeno’s Paradox

That which is in locomo on mustarrive at the half-way stage beforeit arrives at the goal.

(Aristotle Physics VI:9, 239b10)

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Outline

Heuris cs

Errors and tolerances

Examples

Precise Defini on of a Limit

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Heuristic Definition of a LimitDefini onWe 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 to Las we like) by taking x to be sufficiently close to a (on either side ofa) but not equal to a.

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Heuris cs


Examples


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The error-tolerance gameA game between two players (Dana and Emerson) to decide if a limitlimx→a

f(x) exists.

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 tovalues y within the error level of L. If Dana cannot, Emersonwins and the limit cannot be L.

Step 4 If Dana’s move is a good one, Emerson can challenge againor give up. If Emerson gives up, Dana wins and the limit is L.

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The error-tolerance game

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This tolerance is too big

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S ll too big

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This looks good

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So does this

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L

I To be legit, the part of the graph inside the blue (ver cal) stripmust also be inside the green (horizontal) strip.

I Even if Emerson shrinks the error, Dana can s ll move.

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Examples


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Playing the E-T GameExample

Describe how the the Error-Tolerance game would be played todetermine lim

x→0x2.

Solu on

I Dana claims the limit is zero.I If Emerson challenges with an error level of 0.01, Dana needsto guarantee that−0.01 < x2 < 0.01 for all x sufficiently closeto zero.

I If−0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.

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x→0x2.

Solu on

I If Emerson re-challenges with an error level of 0.0001 = 10−4,what should Dana’s tolerance be?

I A tolerance of 0.01 works because|x| < 10−2 =⇒

∣∣x2∣∣ < 10−4.

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x→0x2.

Solu on

I Dana has a shortcut: By se ng tolerance equal to the squareroot of the error, Dana can win every round. Once Emersonrealizes this, Emerson must give up.

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Graphical version of E-T gamewith x2

....x

...

y

No ma er how small anerror Emerson picks,Dana can find a fi ngtolerance band.

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A piecewise-defined functionExample

Find limx→0

|x|x

if it exists.

Solu onThe func on can also be wri en 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 E-T game with a piecewisefunction

Find limx→0

|x|x

if it exists.

.... x...

y

..

−1

..

1

....

.I think the limit is 1

. Can you fit an error of 0.5?.How aboutthis for a tol-erance?

.No. Part ofgraph insideblue is notinside green

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Oh, I guessthe limit isn’t1 .

I think the limitis −1 . Can you fit an

error of 0.5?.

No. Part ofgraph insideblue is notinside green

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Oh, I guessthe limit isn’t−1 .

I think the limitis 0 . Can you fit an

error of 0.5?. No. None of

graph insideblue is insidegreen

.Oh, I guessthe limit isn’t0

.I give up! Iguess there’sno limit!

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One-sided limitsDefini onWe 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 to L aswe like) by taking x to be sufficiently close to a and greater than a.

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One-sided limitsDefini onWe write

limx→a+−

f(x) = L

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 Las we like) by taking x to be sufficiently close to a and less than a.

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Another ExampleExample

Find limx→0+

1xif it exists.

Solu on

The limit does not exist because the func on is unbounded near 0.Next week we will understand the statement that

limx→0+

1x= +∞

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The error-tolerance game with 1/x

Find limx→0+

1xif it exists.

.. x.

y

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L?

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The graph escapesthe green, so nogood

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Even worse!

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The limit does not existbecause the func on isunbounded near 0

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Weird, wild stuffExample

Find limx→0

sin(πx

)if it exists.

Solu on

I f(x) = 0 when x =1kfor any integer k

I f(x) = 1 when x =2

4k+ 1for any integer k

I f(x) = −1 when x =2

4k− 1for any integer k

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Function valuesx π/x sin(π/x)1 π

0

1/2 2π

0

1/k kπ

0

2 π/2

1

2/5 5π/2

1

2/9 9π/2

1

2/13 13π/2

1

2/3 3π/2

− 1

2/7 7π/2

− 1

2/11 11π/2

− 1

...

π/2

..π ..

3π/2

.. 0

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What could go wrong?Summary of Limit Pathologies

How could a func on fail to have a limit? Some possibili es:I le - and right- hand limits exist but are not equalI The func on is unbounded near aI Oscilla on with increasingly high frequency near a

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Meet the MathematicianAugustin Louis Cauchy

I French, 1789–1857I Royalist and CatholicI made contribu ons in geometry,calculus, complex analysis,number theory

I created the defini on of limitwe use today but didn’tunderstand it

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Outline

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Examples


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Precise Definition of a LimitLet f be a func on defined on an some open interval that containsthe number a, except possibly at a itself. Then we say that the limitof 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

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L− ε

.a− δ

.a+ δ

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This δ is too big

.a− δ

.a+ δ

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This δ looks good

.a− δ

.a+ δ

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So does this δ

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L

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SummaryMany perspectives on limits

I Graphical: L is the value the func on“wants to go to” near a

I Heuris cal: f(x) can be made arbitrarilyclose to L by taking x sufficiently closeto a.

I Informal: the error/tolerance gameI Precise: if for every ε > 0 there is acorresponding δ > 0 such that if0 < |x− a| < δ, then |f(x)− L| < ε.

I Algebraic: next me

.. x.

y

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−1

..

1

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FAIL

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lesson 3: the limit of a function (handout)

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