Need help badly... with math

[Deadly]

Seems like a strange place to ask... but what the heck...

I don't suppose anyone around here is well versed in mathematics? Specifically analysis/(multi-variable) calculus. Epsilon-delta definitions, Riemann integrals... and such.

I am in the unfortunate position of having two assignments that I need to get done, and I just can't seem to figure them out. I could ask around at the university on monday, hope someone isn't too busy to help, but I would rather not have to make everything at the last moment unless forced.

What I need is someone to bounce some questions off of, someone to give me a few hints on the problems so that I can hopefully figure them out. Not asking for solutions, just hints. I do enjoy solving these problems myself, otherwise I wouldn't be studying math in the first place, I just can't seem to get the right idea to get me started with some of these problems.

I would prefer PM, if anyone has the time to help out.

[Thufir]

Seems like a strange place to ask... but what the heck...

Strange? We seem to get about one of these threads every fortnight or so.
Well, except most people just post their problems in thread and everyone chips in, rather than asking for PM help.

On the subject of which, I probably could help, but I need to be doing my own assignments/MMath project. I'm sure one of the others will be along presently.

[Deadly]

Strange? We seem to get about one of these threads every fortnight or so.

That makes it less strange? :smalltongue:

And no worries... good luck with your own work.

[Tirian]

Sure, you can PM me if you like.

[Deadly]

Sure, you can PM me if you like.

Done. And thanks in advance :)

[Deathslayer7]

Gimme! Gimme! Gimme! Moar! :smallbiggrin:

[Deadly]

Gimme! Gimme! Gimme! Moar! :smallbiggrin:

Whoa :) I guess I could forward the questions to you as well. Can't hurt with another pair of eyes

[Deadly]

I guess I should just post it here. I'm pretty sure I am allowed to discuss the problems with fellow students and such, so I don't see much of a problem. Just remember I'm not asking for a full solution, just a good hint to get me on the right path.

Anyway:


Hi. Here's the first Q:


Hi. Thanks for letting me bother you. I guess I should just fire away?

One of the smaller ones to start with...

I have an expression:

[x(e^x+1)-2(e^x-1)]/x^3

I need to prove that it has a limit for x->0. Assuming it has one I can find it easily enough, no problem, but I'm asked to prove it has one not necessarily find it.

I have been trying to go through with the epsilon-delta definition, and so I start with

|f(x)| < epsilon where f(x) is the above expression

I pretty much get stuck here. I can't seem to get any useful delta out of that. I've been staring at the definition, and flipping through the book for some theorem or whatever which may be of use, but can't find anything to get me going on the right path.

I've used L'Hôpital's rule, and found the limit to be 1/6, assuming of course it exists.

[Tirian]

Like I said in PM, L'Hopital's rule (or the "stronger form" as it is described in my ancient calculus book) proves that the limit of f(x)/g(x) exists and is the limit of f'(x)/g'(x), assuming the latter limit exists. So if you got to a rational form where the value of the denominator is non-zero, then you shouldn't have any trouble showing that THAT limit exists and then work your way backward from there.

[Deadly]

I'm not certain, but it may work. I haven't tried, though, particularly because L'Hôpital's rule is not part of this course as far as I can see. I am pretty sure we are expected to use the epsilon-delta definition. Everything else that we have discussed apparently assumes the limit exists and so can't be used to prove it.

[neoseph7]

If I may make a general recomendation for limits (or just function analysis in general), try graphing the function in a TI-89 or proper math program (mathematica, tk solver, maple, etc.). Heck, even MS excel in a pinch. Also, I concur on L'hopitalizing the function for the solution. I recall it not giving the actual limit though, instead just showing that the limit exists.

Edit: Graphed it in Excel. 1/6 does indeed appear to be the limit just on visual inspection (1.6667)

[Deadly]

If I may make a general recomendation for limits (or just function analysis in general), try graphing the function in a TI-89 or proper math program (mathematica, tk solver, maple, etc.). Heck, even MS excel in a pinch. Also, I concur on L'hopitalizing the function for the solution. I recall it not giving the actual limit though, instead just showing that the limit exists.

I've plotted it, and I have found the limit using L'Hôpital. The limit exists and is 1/6, I just can't prove it. L'Hôpital may possibly be used to show that it exists and is 1/6, but if so I think it'll be a long and rather ugly proof, and as I said above I'm not even sure using L'Hôpital would be accepted in this course since it's not part of what we have been discussing. I may have to use it if all else fails, but I would prefer not.

Edit: Maple, TI-89 and even WolframAlpha all agree that the limit is 1/6. But either way, finding the limit is not the problem.

[Tirian]

What do you know about limits regarding the exponential function? The way my textbook is structured, transcendential functions are defined after both differentiation and integration.

For instance, if you happened to know that e^x ~ 1 + x + (x^2)/2 + (x^3)/6 + epsilon, I think the math works out okay.

[RS14]

[x(e^x+1)-2(e^x-1)]/x^3

I need to prove that it has a limit for x->0. Assuming it has one I can find it easily enough, no problem, but I'm asked to prove it has one not necessarily find it.

I have been trying to go through with the epsilon-delta definition, and so I start with

|f(x)| < epsilon where f(x) is the above expression.

Don't you mean that

|f(x)-y_0| < epsilon, for some y_0? (e.g. y_0=1/6)

?

[Zephykinns]

Not to be a moron. But the answer is 42. :D

[Deadly]

What do you know about limits regarding the exponential function? The way my textbook is structured, transcendential functions are defined after both differentiation and integration.

For instance, if you happened to know that e^x ~ 1 + x + (x^2)/2 + (x^3)/6 + epsilon, I think the math works out okay.

Hmm. I don't believe we've had anything about that. We haven't really dealt with specific functions much. Maybe something has been mentioned in a small note or example somewhere, though :smallsigh:


Don't you mean that

|f(x)-y_0| < epsilon, for some y_0? (e.g. y_0=1/6)

?

True, but here y_0 would be zero (the value x approaches), so only f(x) is left. Or am I wrong there?

[RS14]

True, but here y_0 would be zero (the value x approaches), so only f(x) is left. Or am I wrong there?

No, I don't think that's right. The epsilon is a maximum; we assert that we can always find some delta such that, for all x closer to x_0 than delta, f(x) is closer than epsilon to the limit y_0. What is the distance of f(x) from the limit? |f(x)-y_0|. In this case, x_0=0 and y_0 apparently would be 1/6.

[Deadly]

and y_0 apparently would be 1/6.

So... you want to use the limit itself, in the proof of its existence :smallconfused:

Edit: On closer examination it'd seem there may be something to what you say. It just seems wrong to use the limit like this, in its own proof. And either way it doesn't help on the fact that I can't apparently get anywhere from there. I can't find a suitable delta from |f(x)-1/6|<epsilon any more than I could for |f(x)|<epsilon

[drakir_nosslin]

Quick question before I take a closer look at this one. Is it supposed to be
[x(e^x+1)-2(e^x-1)]/x^3
or
[x(e^(x+1))-2(e^(x-1))]/x^3?
It's easy to miss a parenthesis, I do it all the time. :smalltongue:

[Deadly]

Quick question before I take a closer look at this one. Is it supposed to be
[x(e^x+1)-2(e^x-1)]/x^3
or
[x(e^(x+1))-2(e^(x-1))]/x^3?
It's easy to miss a parenthesis, I do it all the time. :smalltongue:

The first one. No missed parentheses in this case :)

[drakir_nosslin]

Ok, good. I'll see what I can come up with. Can't promise anything though.

[Deadly]

Wouldn't ask for promises :smallwink: But thanks regardless

[drakir_nosslin]

Well, I have found one way to prove that it exists, however you find it at the same time :smallbiggrin:. We never used the epsilon-delta definition much, so I skipped that and did like this instead:
f(x)=[x(e^x+1)-2(e^x-1)]/x^3
Let lim x->0 [x(e^x+1)-2(e^x-1)]/x^3 = L

L=A-B, then we can say f(x)=h(x)-g(x) where
h(x)=(e^x+1)/x^2 and
g(x)=2(e^x-1)/x^3

Knowing that e^x=sum from n=0 to inf of x^n/n!=(1+x+x^2/2!+...) we can derive the following:

h(x)=1/x^2(1+x+x^2/2+x^3/3!+...)+1/x^2 =>(1/x^2+1/x+1/2+x/3!+...)+1/x^2 = 2/x^2+1/x+1/2+x/3!+...

g(x)=2/x^3(1+x+x^2/2+x^3/3!+...)-2/x^3 = 2/x^3+2/x^2+1/x+1/3!+...-2/x^3

So h(x)-g(x)= 1/2-1/3!+...-... where every term in the ... approaches zero as x->0. So the limit exists and it should be at 2/6.

Now I know that this isn't the right limit, since the rest of you got 1/6, but I'm not really good at this anyway. Perhaps you can use it for something at least?
Oh, and there's probably tons of errors in the above as well, but I think that the theory holds. Hopefully.

[Deadly]

Clearly some kind of error, since you got 2/6 instead of 1/6. But thanks anyway, I hope I can use it somehow. If not all of it, maybe parts of it will help. I'm starting to get a little desperate.

Edit: Ha! I got it. Apparently I was supposed to use L'Hôpital for it. Or at least a version of L'Hôpital hidden in a part of the book where I didn't expect to find anything related to this problem. Just typical.

Thanks for all the help, all of you. I may post about other problems later, but I think I'll try taking another look at them myself first.

[Tirian]

Hmm. I don't believe we've had anything about that. We haven't really dealt with specific functions much. Maybe something has been mentioned in a small note or example somewhere, though :smallsigh:

Hmm, let me ask this way then. If you were to do a delta-epsilon proof, how would you simplify e^(x+delta_x)? Or even an estimate for e^x for small x in the same way that we say that sin x ~ x for x near zero?

If that's also unfamiliar to you, then this seems like a really cheesy problem, asking you to solve a complex limit without pretending that L'Hopital's Rule or Taylor series exist. You might be forced to ask tomorrow after all to find out what they were expecting of you (and then be sure to tell us about it so we can tell you how elaborate your revenge should be :smallbiggrin:).

ETA: Ah, just noticed that you had found a reference to L'Hopital. Hooray!