Math Is Fun Forum

Anakin

Member

Registered: 2009-10-04

Posts: 145

Proving Limits

Ooh, it's been quite a while since I've been here. I'd stay around more if I had much to contribute.

Anyway, hello everyone. I had some questions that I needed some help with. There are 5 of them and I've completed all but the 4th one. But I'd like confirmation for my solution for #1 and #5.

Here are the questions: http://i56.tinypic.com/30nb1xf.png

Question 1 solution: http://i52.tinypic.com/2hfht83.jpg
Question 5 solution: http://i56.tinypic.com/2jdhlzl.jpg
Question 4 beginning: http://i53.tinypic.com/15dvs6w.jpg

Number 2 and 3 I am confident in. If anyone could look over the work that I've done with the 1st and 5th questions and provide some guidance on question 4, I'd be very thankful.

Anakin

Member

Registered: 2009-10-04

Posts: 145

Re: Proving Limits

Hi again,

With much aid from someone else online, I was able to get the following solution for Q4. Does it look about right?

http://i52.tinypic.com/9itu6p.jpg

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Proving Limits

Hi Anakin,

I haven't solved limits in this manner. I need to look around.
What is it called?

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Proving Limits

Hi Anakin;

Good to see you. I would love to help you but I am drawing a blank on it since you first posted it. Sorry but I have nothing to add.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Anakin

Member

Registered: 2009-10-04

Posts: 145

Re: Proving Limits

Hi gAr and hi Bobbym, indeed it is!

This is using the δ-ϵ (delta-epsilon) definition to prove limits. Then there is the N-ϵ method to prove limits which involve infinity.

So for the δ-ϵ method, it is basically saying the following: if we have lim of x -> a f(x) = L. For all x, if 0 < |x-a| < δ (we choose this δ), then |f(x) - L| < ϵ.

We take ϵ > 0 to be arbitrary and then prove the implication (the then statement). If the proof is successful, then the limit has been proven. The N-ϵ is similar but a bit different. Here is the Wikipedia article on them (I'm surprised you guys have never come across this, perhaps it isn't used much outside of low-level Calculus): http://en.wikipedia.org/wiki/%28%CE%B5,_%CE%B4%29-definition_of_limit

Last edited by Anakin (2011-06-10 21:10:17)

reconsideryouranswer

Member

Registered: 2011-05-11

Posts: 171

Re: Proving Limits

Anakin, you typed steps where you have this variable constant, a,
but you did not state where it came from and/or what classification
of numbers it belongs to.

It just shows up without being introduced.

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Anakin

Member

Registered: 2009-10-04

Posts: 145

Re: Proving Limits

Hi Reconsideryouranswer,

If you are referring to my last post, a is a real number. As in as x is approaching some real number a, f(x) approaches L, some other real number (or the limit).

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Proving Limits

Hi Anakin,

Okay, thanks for explaining.
Good to know that you got your answer right.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Anakin

Member

Registered: 2009-10-04

Posts: 145

Re: Proving Limits

Hi gAr,

I'm not actually certainly sure that my Q1 is right as it uses a one sided limit and it uses negative infinity, but I think it is by using the following reasoning:

Let N < 0 be arbitrary, and thus it is negative which makes sense as the limit is -∞.

We choose δ = 2^N.

If 0 < |x-3| < δ:

<=> -δ < x-3 < δ
<=>  -δ < 3-x < δ (by multiplying each side by -1 and thus signs reverse)
=> 3-x < 2^N (δ = 2^N as chosen)
=> log(3-x) with base 2 < N (as 3-x > 0 since x is approaching 3 from the left side and the base >= 1 so the sign doesn't change).

Therefore, the function log(3-x) with base 2 has a limit of -∞ as it is always less than N when approaching 3 from the left side.

End of proof.

Last edited by Anakin (2011-06-10 21:58:58)

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Proving Limits

Hi Anakin,

I finally found an example which is similar to the one you wanted!

http://math.gcsu.edu/~ryan/1261/notes/limit2.pdf
Pages 3 and 4, hope it helps!

Last edited by gAr (2011-06-11 04:52:43)

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Anakin

Member

Registered: 2009-10-04

Posts: 145

Re: Proving Limits

Hi gAr,

I appreciate you taking the time to find that link. Apparently my method was wrong (according to that source) and after some thinking, rightfully so. I've made the adjustments and I'm now confident in my answers.

Thanks once again.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Proving Limits

Hi Anakin,

You're welcome.
Happy learning!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."