lesson 3: limits (section 21 slides)
TRANSCRIPT
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.
Section 1.3The Concept of Limit
V63.0121.021, Calculus I
New York University
September 14, 2010
Announcements
I Let us know if you bought a WebAssign license last year andcannot login
I First written HW due ThursdayI Get-to-know-you survey and photo deadline is October 1
. . . . . .
. . . . . .
Announcements
I Let us know if you boughta WebAssign license lastyear and cannot login
I First written HW dueThursday
I Get-to-know-you surveyand photo deadline isOctober 1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 2 / 39
. . . . . .
Guidelines for written homework
I Papers should be neat and legible. (Use scratch paper.)I Label with name, lecture number (021), recitation number, date,
assignment number, book sections.I Explain your work and your reasoning in your own words. Use
complete English sentences.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 3 / 39
. . . . . .
Rubric
Points Description of Work3 Work is completely accurate and essentially 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 ex-plained, 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 ofwork is incorrect.
0 Work minimal or non-existent. Solution is completely in-correct.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 4 / 39
. . . . . .
Examples of written homework: Don't
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 5 / 39
. . . . . .
Examples of written homework: Do
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 6 / 39
. . . . . .
Examples of written homework: DoWritten Explanations
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 7 / 39
. . . . . .
Examples of written homework: DoGraphs
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 8 / 39
. . . . . .
Objectives
I Understand and state theinformal definition of a limit.
I Observe limits on a graph.I Guess limits by algebraic
manipulation.I Guess limits by numerical
information.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 9 / 39
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.
Limit
. . . . . .
. . . . . .
Yoda on teaching a concepts course
“You must unlearn what you have learned.”
In other words, we are building up concepts and allowing ourselvesonly to speak in terms of what we personally have produced.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 11 / 39
. . . . . .
Zeno's Paradox
That which is inlocomotion must arriveat the half-way stagebefore it arrives at thegoal.
(Aristotle Physics VI:9, 239b10)
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 12 / 39
. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 13 / 39
. . . . . .
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 to L aswe like) by taking x to be sufficiently close to a (on either side of a) butnot equal to a.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 14 / 39
. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 15 / 39
. . . . . .
The error-tolerance game
A 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 counting 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 again orgive up. If Emerson gives up, Dana wins and the limit is L.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 16 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big
.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big
.Still too big
.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big
.This looks good
.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good
.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I Even if Emerson shrinks the error, Dana can still move.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 18 / 39
. . . . . .
Example
Find limx→0
x2 if it exists.
Solution
I Dana claims the limit is zero.I 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 tozero.
I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.I If Emerson re-challenges with an error level of 0.0001 = 10−4,
what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒
∣∣∣x2∣∣∣ < 10−4.I Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
. . . . . .
Example
Find limx→0
x2 if it exists.
Solution
I Dana claims the limit is zero.
I If Emerson challenges with an error level of 0.01, Dana needs toguarantee that −0.01 < x2 < 0.01 for all x sufficiently close tozero.
I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.I If Emerson re-challenges with an error level of 0.0001 = 10−4,
what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒
∣∣∣x2∣∣∣ < 10−4.I Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
. . . . . .
Example
Find limx→0
x2 if it exists.
Solution
I Dana claims the limit is zero.I 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 tozero.
I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.I If Emerson re-challenges with an error level of 0.0001 = 10−4,
what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒
∣∣∣x2∣∣∣ < 10−4.I Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
. . . . . .
Example
Find limx→0
x2 if it exists.
Solution
I Dana claims the limit is zero.I 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 tozero.
I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
I If Emerson re-challenges with an error level of 0.0001 = 10−4,what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒
∣∣∣x2∣∣∣ < 10−4.I Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
. . . . . .
Example
Find limx→0
x2 if it exists.
Solution
I Dana claims the limit is zero.I 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 tozero.
I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.I If Emerson re-challenges with an error level of 0.0001 = 10−4,
what should Dana’s tolerance be?
A tolerance of 0.01 worksbecause |x| < 10−2 =⇒
∣∣∣x2∣∣∣ < 10−4.I Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
. . . . . .
Example
Find limx→0
x2 if it exists.
Solution
I Dana claims the limit is zero.I 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 tozero.
I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.I If Emerson re-challenges with an error level of 0.0001 = 10−4,
what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒
∣∣∣x2∣∣∣ < 10−4.
I Dana has a shortcut: By setting tolerance equal to the square rootof the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
. . . . . .
Example
Find limx→0
x2 if it exists.
Solution
I Dana claims the limit is zero.I 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 tozero.
I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.I If Emerson re-challenges with an error level of 0.0001 = 10−4,
what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒
∣∣∣x2∣∣∣ < 10−4.I Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error Emerson picks, Dana can find afitting tolerance band.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
. . . . . .
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
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 21 / 39
. . . . . .
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
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 21 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
The E-T game with a piecewise function
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 about thisfor a tolerance?.
No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t 1
.I think the limit is−1
.Can you fit anerror of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess thelimit isn’t −1
.I think the limit is 0
.Can you fit anerror of 0.5?.
No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
. . . . . .
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 to L aswe like) by taking x to be sufficiently close to a and greater than a.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 23 / 39
. . . . . .
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 to L aswe like) by taking x to be sufficiently close to a and less than a.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 23 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
..Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
The error-tolerance game
Find limx→0+
|x|x
and limx→0−
|x|x
if they exist.
. .x
.y
..−1
..1 .
.
.Part of graphinside blue isinside green
.Part of graphinside blue isinside green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
. . . . . .
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
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 25 / 39
. . . . . .
Example
Find limx→0+
1xif it exists.
SolutionThe limit does not exist because the function is unbounded near 0.Next week we will understand the statement that
limx→0+
1x= +∞
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 26 / 39
. . . . . .
The error-tolerance game
Find limx→0+
1xif it exists.
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not exist be-cause the function is un-bounded near 0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
. . . . . .
The error-tolerance game
Find limx→0+
1xif it exists.
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not exist be-cause the function is un-bounded near 0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
. . . . . .
The error-tolerance game
Find limx→0+
1xif it exists.
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not exist be-cause the function is un-bounded near 0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
. . . . . .
The error-tolerance game
Find limx→0+
1xif it exists.
. .x
.y
.0
..L?
.The graph escapesthe green, so no good
.Even worse!
.The limit does not exist be-cause the function is un-bounded near 0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
. . . . . .
The error-tolerance game
Find limx→0+
1xif it exists.
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not exist be-cause the function is un-bounded near 0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
. . . . . .
The error-tolerance game
Find limx→0+
1xif it exists.
. .x
.y
.0
..L?
.The graph escapesthe green, so no good
.Even worse!
.The limit does not exist be-cause the function is un-bounded near 0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
. . . . . .
The error-tolerance game
Find limx→0+
1xif it exists.
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not exist be-cause the function is un-bounded near 0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
. . . . . .
Example
Find limx→0+
1xif it exists.
SolutionThe limit does not exist because the function is unbounded near 0.Next week we will understand the statement that
limx→0+
1x= +∞
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 28 / 39
. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =
1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 29 / 39
. . . . . .
Function values
x π/x sin(π/x)1 π 0
1/2 2π 01/k kπ 02 π/2 1
2/5 5π/2 12/9 9π/2 12/13 13π/2 12/3 3π/2 −12/7 7π/2 −12/11 11π/2 −1
.
..π/2
..π
..3π/2
. .0
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 30 / 39
. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =
1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =
1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
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 =
24k− 1
for any integer k
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
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
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
. . . . . .
Weird, wild stuff continued
Here is a graph of the function:
. .x
.y
..−1
..1
There are infinitely many points arbitrarily close to zero where f(x) is 0,or 1, or −1. So the limit cannot exist.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 32 / 39
. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 33 / 39
. . . . . .
What could go wrong?Summary of Limit Pathologies
How could a function fail to have a limit? Some possibilities:I left- and right- hand limits exist but are not equalI The function is unbounded near aI Oscillation with increasingly high frequency near a
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 34 / 39
. . . . . .
Meet the Mathematician: Augustin Louis Cauchy
I French, 1789–1857I Royalist and CatholicI made contributions in
geometry, calculus,complex analysis, numbertheory
I created the definition oflimit we use today butdidn’t understand it
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 35 / 39
. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 36 / 39
. . . . . .
Precise Definition of a LimitNo, this is not going to be on the test
Let f be a function defined on an some open interval that contains thenumber a, except possibly at a itself. Then we say that the limit 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.
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 37 / 39
. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
. . . . . .
Summary: Many perspectives on limits
I Graphical: L is the valuethe function “wants to goto” near a
I Heuristical: f(x) can bemade arbitrarily close to Lby taking x sufficientlyclose to a.
I Informal: theerror/tolerance game
I Precise: if for every ε > 0there is a correspondingδ > 0 such that if0 < |x− a| < δ, then|f(x)− L| < ε.
I Algebraic: next time
. .x
.y
..−1
..1
.FAILV63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 39 / 39