Limits

Frodo Baggins · 07-14-04, 02:03 PM

Can someone please check this? I have my doubts, and I need to be solid at this stuff.

Prove lim_(x -> 3-) = 2(x+3)/|x-3| = -2 = L

Where that limit reads as x approaches 3 from the left.
I'll use f(x) = 2(x + 3)/|x-3|

Note,
|f(x) - L | = | 2(x + 3)/|x - 3| + 2 | = |2(x + 3) + 2(x - 3) | / |x - 3| = 2 |(x - 3) + 2|x - 3||/ |x - 3|

But if |x - 3| < 1, then -1 < x - 3 < 1 and x - 3 < 1

So, we are left with,
|f(x) - L| < 2 | 1 + 2|x - 3|| / 1 = 2 + 4|x - 3|

Thus, let e > 0 be arbitrary. Let d = min{1, (e - 2) / 4}. Then if 3 - d < x < 3, then, |x - 3| < (e - 2)/4

|f(x) - L| = 2 |(x - 3) + 2|x - 3||/ |x - 3| < 2 + 4|x - 3| < 2 + (e - 2)/4 = e

QED

cw823 · 07-14-04, 02:58 PM

Quote:

Originally Posted by Frodo Baggins

Can someone please check this? I have my doubts, and I need to be solid at this stuff.

Prove lim_(x -> 3-) = 2(x+3)/|x-3| = -2 = L

Where that limit reads as x approaches 3 from the left.
I'll use f(x) = 2(x + 3)/|x-3|

Note,
|f(x) - L | = | 2(x + 3)/|x - 3| + 2 | = |2(x + 3) + 2(x - 3) | / |x - 3| = 2 |(x - 3) + 2|x - 3||/ |x - 3|

But if |x - 3| < 1, then -1 < x - 3 < 1 and x - 3 < 1

So, we are left with,
|f(x) - L| < 2 | 1 + 2|x - 3|| / 1 = 2 + 4|x - 3|

Thus, let e > 0 be arbitrary. Let d = min{1, (e - 2) / 4}. Then if 3 - d < x < 3, then, |x - 3| < (e - 2)/4

|f(x) - L| = 2 |(x - 3) + 2|x - 3||/ |x - 3| < 2 + 4|x - 3| < 2 + (e - 2)/4 = e

QED

It appears to be very boring.

Cerberus2k7 · 07-14-04, 05:11 PM

*Walks into thread*

*Sees very very very very confusing lines and numbers*

*Runs out screaming something about badgers*

deRusett · 07-14-04, 10:28 PM

Wantto write it out and post an image of it, I'm not making sence of it, because I have never seen Limits done typed on single lines, so its a bit confusing,

side note.
Want a good textbook for this and related items
Pick up
Essentials of Technical Mathematics with Calculus second Edition
by Richard Paul/M.Leonard Shaevel
ISBN 01-13-289091-7

its an older book it was my First Year Math text book, but has become my book of referance when ever I need a refersher on this stuff

Oni · 07-14-04, 10:36 PM

Quote:

Originally Posted by cw823

It appears to be very boring.

You would appear to be right

deRusett · 07-14-04, 11:00 PM

thats why they discovered Differentiation

because limits are very boring, but you need to understand them before you can understand what a Derivative actually is.

then you get to forget how to take limits

thats a happy day

shard · 07-15-04, 12:00 AM

*walks around lost and feeling stupid*
*reads calculus book and head explodes*
*whaaaaaaaaaaaaaaaaaaa he cries me stupid*

JigPu · 07-15-04, 12:13 AM

Woah... Way too many parentathes and absolute value signs. My head is officially spinning now. I think the lack of knowing what's being divided by what is really throwing me off. Any way you could put up a copy that's not on one line?

Quote:

Originally Posted by deRusett

thats why they discovered Differentiation

because limits are very boring, but you need to understand them before you can understand what a Derivative actually is.

then you get to forget how to take limits

thats a happy day

The day I forgot how to take a limit (ok.. so it hasn't happened yet, but I've been close) was the worst day of my life! Reducing calculus to just sets of rules and regulations denies it of the mathematical beauty that lies underneath. While I think limits are evil (I prefer just plugging infinity/infintessimal straight into the equation

), the idea behind them is interesting and thought provoking.

...though the rules and regulations do make it a LOT easier to blaze through the zillion problems assigned!

JigPu

Drec · 07-15-04, 12:24 AM

*pretends like he didn't see this thread*

*walks away slowly*

SK8 · 07-15-04, 12:31 AM

wow do i got somthing to look forward to in a few years! Algebra 2 next year :P

zexmarquies01 · 07-15-04, 01:01 AM

i can't even read that thing at all, and my head hurts just by looking at it!!!!

Shard, we are in the same boat. me is very stupid too.

This hurts my head to much, i think i'm going to go look at some Pron now....less thinking required.

Frodo Baggins · 07-15-04, 08:36 AM

Quote:

Originally Posted by deRusett

Wantto write it out and post an image of it, I'm not making sence of it, because I have never seen Limits done typed on single lines, so its a bit confusing,

germanjulian · 07-15-04, 08:45 AM

***** whoo what*******
*****tries to remember arithmetic skills aquired in school -> fail*****
*****tries to disable dyslexia and enable spelling skills -> fail *****
*****tries to understand theories about quantum physics, relativity, super string theory etc -> yeah makes sense just dont get the math******
***** lets go back to php programing*****

sandman001 · 07-15-04, 12:08 PM

I don't have a clue, but I give anyone who understands this some mad props.

Roboman101 · 07-15-04, 12:08 PM

from what i remember from earlier this year in calc I, and some of calc II, that seems to be done more in depth than i remember doing epsilon and delta proofs when i had the unpleasant experience of doing them

as i recall, you really don't need to know how to do these that much, just remeber the limits and how to take them and your set

i get the limit to be 1.2 x 10^14th

that is on an ti-86 doing it graphically and via table, and by hand getting something big

i'd say your limit should be either infinity or something along the lines of the above

deRusett · 07-15-04, 12:14 PM

question1, who said the Lim as X aproches 3 from the left is -2? I get it to be 12 from the equation

questio2
Why did you bother with this line
"But if |x-3| <1, then -1 < x -3 < 1 and x+3 < 7

X+3 can not equal 7 because we ignore the function after X=3

as we aproche 3 from the left we get x=2.99999999999999
which makes |x-3| = 0.0000000000001 ( I didn't count the 9's or zeros just held my finger down )

I must be missing something because unless x is aproching -3 from the left there is no way it can be a negative value

I'd go into more detail but I have 5 minutes to get ready for work, and I don't want to break out a pen and paper

macky · 07-15-04, 02:14 PM

When did they start using letters in math?

Docta_Z · 07-15-04, 02:29 PM

You are making life hard for yourself on the third line in as deRusett says.

No need to rearrange it to test at 1.

But looking back - is your whole first line given in the question?

I don't think it equals 2...

I did this stuff 2 years ago, now all my calculus is completely different. I'll think about it for a while though.

Quote:

Originally Posted by macky

When did they start using letters in math?

Grade 8 for me...

David · 07-15-04, 02:32 PM

I should probably know the answer to this.

I should probably be able to give some input.

I should have actually gone to more than half my maths lectures last year.

Damn.

Frodo Baggins · 07-15-04, 03:07 PM

Quote:

Originally Posted by deRusett

question1, who said the Lim as X aproches 3 from the left is -2? I get it to be 12 from the equation

questio2
Why did you bother with this line
"But if |x-3| <1, then -1 < x -3 < 1 and x+3 < 7

X+3 can not equal 7 because we ignore the function after X=3

as we aproche 3 from the left we get x=2.99999999999999
which makes |x-3| = 0.0000000000001 ( I didn't count the 9's or zeros just held my finger down )

I must be missing something because unless x is aproching -3 from the left there is no way it can be a negative value

I'd go into more detail but I have 5 minutes to get ready for work, and I don't want to break out a pen and paper

I didn't mean to waste your time. The equation is written wrong. I'll fix some stuff up

Quote:

questio2
Why did you bother with this line
"But if |x-3| <1, then -1 < x -3 < 1 and x+3 < 7

X+3 can not equal 7 because we ignore the function after X=3

It's a bit obsolete now that the equation is wrong. But I never said it was equal to 7.
-1 < x - 3 < 1 => -1 + 6 < x + 3 < 1 + 6 => 5 < x + 3 < 7

We don't worry about the lower bound since -1 < x - 3 < 1 => x + 3 < 7

The reason why was to bound the factor in the expression by a number.

Okay, another try. I've been feeling under the weather, but I hope I didn't commit any serious errors in this one either.

07-14-04, 02:03 PM	Thread Starter #1
Frodo Baggins Member Join Date: Aug 2001 Location: City of Dreaming Spires	Limits Can someone please check this? I have my doubts, and I need to be solid at this stuff. Prove lim_(x -> 3-) = 2(x+3)/\|x-3\| = -2 = L Where that limit reads as x approaches 3 from the left. I'll use f(x) = 2(x + 3)/\|x-3\| Note, \|f(x) - L \| = \| 2(x + 3)/\|x - 3\| + 2 \| = \|2(x + 3) + 2(x - 3) \| / \|x - 3\| = 2 \|(x - 3) + 2\|x - 3\|\|/ \|x - 3\| But if \|x - 3\| < 1, then -1 < x - 3 < 1 and x - 3 < 1 So, we are left with, \|f(x) - L\| < 2 \| 1 + 2\|x - 3\|\| / 1 = 2 + 4\|x - 3\| Thus, let e > 0 be arbitrary. Let d = min{1, (e - 2) / 4}. Then if 3 - d < x < 3, then, \|x - 3\| < (e - 2)/4 \|f(x) - L\| = 2 \|(x - 3) + 2\|x - 3\|\|/ \|x - 3\| < 2 + 4\|x - 3\| < 2 + (e - 2)/4 = e QED
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07-14-04, 05:11 PM	#3
Cerberus2k7 Drifto Mexicano Join Date: Feb 2004 Location: Denver, CO	Walks into thread Sees very very very very confusing lines and numbers Runs out screaming something about badgers __________________ Member of OCdriftingNinjas \| You gotta' Fold if you wanna turn up the Heat Overclocking: My Anti-Drug. \| Overclocking my body...one day at a time... \| Black Belt Ubercloxx0r \| You are the most ripped yo-yo'er I have ever seen. - Seven HTPC \| i5 2500k@3.8 \| Gigabyte Z68X-UD3H-B3 \| 16GB HyperX \| GTX 460 1gb@845/1100 \| 1.5TB \| Silverstone LC10S-M \| Corsair 500w
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07-14-04, 10:28 PM	#4
deRusett Member Join Date: Jul 2002 Location: Midland, Ontario	Wantto write it out and post an image of it, I'm not making sence of it, because I have never seen Limits done typed on single lines, so its a bit confusing, side note. Want a good textbook for this and related items Pick up Essentials of Technical Mathematics with Calculus second Edition by Richard Paul/M.Leonard Shaevel ISBN 01-13-289091-7 its an older book it was my First Year Math text book, but has become my book of referance when ever I need a refersher on this stuff __________________ Success is.. At age 4 .... success is .... not peeing in your pants. At age 12 .... success is .... having friends. At age 17 .... success is .... having a driving licence. At age 20 .... success is .... having sex. At age 35 .... success is .... having money. At age 50 .... success is .... having money. At age 60 .... success is .... having sex. At age 70 .... success is .... having a driving licence. At age 75 .... success is .... having friends. At age 80 .... success is .... not peeing in your pants At Any Age ... success is ... Folding for Team 32
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07-14-04, 11:00 PM	#6
deRusett Member Join Date: Jul 2002 Location: Midland, Ontario	thats why they discovered Differentiation because limits are very boring, but you need to understand them before you can understand what a Derivative actually is. then you get to forget how to take limits thats a happy day __________________ Success is.. At age 4 .... success is .... not peeing in your pants. At age 12 .... success is .... having friends. At age 17 .... success is .... having a driving licence. At age 20 .... success is .... having sex. At age 35 .... success is .... having money. At age 50 .... success is .... having money. At age 60 .... success is .... having sex. At age 70 .... success is .... having a driving licence. At age 75 .... success is .... having friends. At age 80 .... success is .... not peeing in your pants At Any Age ... success is ... Folding for Team 32
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07-15-04, 12:00 AM	#7
shard Member Join Date: Sep 2003 Location: You Should Make a Wish!	walks around lost and feeling stupid reads calculus book and head explodes whaaaaaaaaaaaaaaaaaaa he cries me stupid __________________ Intel i5 2500k @ 4.6ghz X-Mystique DDL Surround Soundcard 7.1 Harman Kardon AVR 135 \| JBL E30 Pair \| BIC F3 Center \| Dayton Reference 12" 4cu ft box Gigabyte Z77-D3H G. Skill DD3 1600 8gb EVGA 670 GTX \| Asus VE248H 24" Widescreen LCD Corsair 750TX 750w PSU Gettin' da Heat on mon!
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07-15-04, 12:24 AM	#9
Drec Member Join Date: May 2004	pretends like he didn't see this thread walks away slowly
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07-15-04, 12:31 AM	#10
SK8 Member Join Date: Dec 2003 Location: Fort Wayne , In	wow do i got somthing to look forward to in a few years! Algebra 2 next year :P
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07-15-04, 01:01 AM	#11
zexmarquies01 Join Date: May 2002 Location: Ohio	i can't even read that thing at all, and my head hurts just by looking at it!!!! Shard, we are in the same boat. me is very stupid too. This hurts my head to much, i think i'm going to go look at some Pron now....less thinking required. __________________ AMD X6 1055T @3.8Ghz GIGABYTE 890FX-A UD5 6 Gigs of G Skill DDR3 1600 RAM EVGA GTX 460 HAF 932 Tower "Talking about how fast you beat a game is like talking about how fast you are in bed." Do you like Anime? Then join the OCF anime club! OCF Anime Club Click here, and click JOIN!
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07-15-04, 08:45 AM	#13
germanjulian Member Join Date: Apr 2002 Location: Frankfurt/London	*** whoo what*** *tries to remember arithmetic skills aquired in school -> fail* *tries to disable dyslexia and enable spelling skills -> fail * *tries to understand theories about quantum physics, relativity, super string theory etc -> yeah makes sense just dont get the math** * lets go back to php programing*** __________________ /\|\ Asus P5W DH Deluxe, Intel C2D E6600, 4GB Corsair XMS2-6400C4 DDR2, E-VGA GeForce 7800 GT, Asus Xonar D2, 160GB Intel X25-m SSD, 1TB HD, Coolermaster Cosmos, etc. see my website google my name/\|\
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07-15-04, 12:08 PM	#14
sandman001 Just Freeze It Join Date: Mar 2003	I don't have a clue, but I give anyone who understands this some mad props.
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07-15-04, 12:08 PM	#15
Roboman101 Member Join Date: Oct 2002 Location: south dakota	from what i remember from earlier this year in calc I, and some of calc II, that seems to be done more in depth than i remember doing epsilon and delta proofs when i had the unpleasant experience of doing them as i recall, you really don't need to know how to do these that much, just remeber the limits and how to take them and your set i get the limit to be 1.2 x 10^14th that is on an ti-86 doing it graphically and via table, and by hand getting something big i'd say your limit should be either infinity or something along the lines of the above Last edited by Roboman101; 07-15-04 at 12:19 PM. Reason: got the limit of f(x)
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07-15-04, 12:14 PM	#16
deRusett Member Join Date: Jul 2002 Location: Midland, Ontario	question1, who said the Lim as X aproches 3 from the left is -2? I get it to be 12 from the equation questio2 Why did you bother with this line "But if \|x-3\| <1, then -1 < x -3 < 1 and x+3 < 7 X+3 can not equal 7 because we ignore the function after X=3 as we aproche 3 from the left we get x=2.99999999999999 which makes \|x-3\| = 0.0000000000001 ( I didn't count the 9's or zeros just held my finger down ) I must be missing something because unless x is aproching -3 from the left there is no way it can be a negative value I'd go into more detail but I have 5 minutes to get ready for work, and I don't want to break out a pen and paper __________________ Success is.. At age 4 .... success is .... not peeing in your pants. At age 12 .... success is .... having friends. At age 17 .... success is .... having a driving licence. At age 20 .... success is .... having sex. At age 35 .... success is .... having money. At age 50 .... success is .... having money. At age 60 .... success is .... having sex. At age 70 .... success is .... having a driving licence. At age 75 .... success is .... having friends. At age 80 .... success is .... not peeing in your pants At Any Age ... success is ... Folding for Team 32
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07-15-04, 02:14 PM	#17
macky Macaddict Join Date: May 2001 Location: Omaha, NE	When did they start using letters in math?
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07-15-04, 02:32 PM	#19
David Forums Super Moderator Overclockers.com Lead Editor Join Date: Feb 2001 Location: Hiding from Americans and making pretty colours in a drybox in St Andrews	I should probably know the answer to this. I should probably be able to give some input. I should have actually gone to more than half my maths lectures last year. Damn. __________________ David J. Nelson MChem(Edin) PhD(Strath) AMRSC [Academia Profile] OC Forums Super Moderator // Overclockers.com Editor The Workhorse: [Lenovo W510] Intel Core i7-720QM / 8 GB DDR3 / nVidia Quadro FX880M / Crucial M4 256 GB SSD / Windows 7 The HTPC/Server: AMD Phenom X4 9650 / 6 GB DDR2 / nVidia 8300 / 5 TB of HDDs / Antec Solo II Case / Windows 7 The Gaming Rig: AMD Llano A6-3650 / 4 GB DDR3 / nVidia GTX260 / 2x 500 GB HDD in RAID0 / Antec Dark Fleet DF-85 Case / Windows 7 The Benchmarking Setup: Various LGA775 chips / Asus Rampage Formula / 2 GB OCZ 1066 MHz DDR2 / nVidia Quadro NVS 285 / 320 GB HDD / Windows 7 My research fellowship is eating all my time, so I may be slow to reply to emails and private messages.
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