Bike Forums > Foo > Proving limit statement --- Got delta, now what?
 Register All Albums Elite Membership Forum Rules Calendar Search Today's Posts Mark Forums Read

 Foo Off-Topic chit chat with no general subject.

 01-26-09, 09:28 PM #1 phantomcow2 la vache fantôme Thread Starter     Join Date: Aug 2004 Location: NH Bikes: Posts: 6,266 Mentioned: 0 Post(s) Tagged: 0 Thread(s) Quoted: 0 Post(s) Proving limit statement --- Got delta, now what? I'm missing something here. I can find out what to use for delta, but how do I proceed to prove a limit statement to be correct? For example: let e = epsilon and d = delta Lim (x/5) as x -->3 = 3/5 =|(x/5) - 3/5| =|(x-3)/5| Combine terms =.2|x-3| Factor out 1/5 =.2|x-3| < e =|x-3| < 5e Divide both sides by .2 So I know I am supposed to use 5e, right? Aren't I supposed to somehow prove that: if 0 < |x-3| < e then |(x/5)-3/5| < e I'm not really sure how to go about this. Can someone shed some light on the procedure? __________________ C://dos C://dos.run run.dos.run
 01-26-09, 09:29 PM #2 phantomcow2 la vache fantôme Thread Starter     Join Date: Aug 2004 Location: NH Bikes: Posts: 6,266 Mentioned: 0 Post(s) Tagged: 0 Thread(s) Quoted: 0 Post(s) I feel like I am almost there but not quite. Ironically, sort of like a limit. __________________ C://dos C://dos.run run.dos.run
 01-26-09, 09:36 PM #3 jschen riding once again     Join Date: Oct 2005 Location: San Diego, CA Bikes: '06 Cervelo R3, '05 Specialized Allez Posts: 7,359 Mentioned: 0 Post(s) Tagged: 0 Thread(s) Quoted: 0 Post(s) Huh? I lost you after: Lim (x/5) as x --> 3 = 3/5 You prove that a function reaches a limit by demonstrating that an arbitrarily small difference from the limiting value of the input (say, a deviation of d) results in an arbitrarily small change from the proposed limiting value in the function output (say, a difference of e). I assume you're trying to prove that Lim (x/5) as x --> 3 = 3/5. If so, how do your "equations" as they're set up do anything of that sort? Where is the arbitrarily small difference in the input value? Actually, I'm not even sure what your equation is since I don't see = anywhere. But since I don't see anything involving arbitrarily small values, it's not particularly promising. Does this shed any light on why what you've written so far doesn't make much sense conceptually? __________________ If you notice this notice then you will notice that this notice is not worth noticing.
 01-26-09, 09:46 PM #4 phantomcow2 la vache fantôme Thread Starter     Join Date: Aug 2004 Location: NH Bikes: Posts: 6,266 Mentioned: 0 Post(s) Tagged: 0 Thread(s) Quoted: 0 Post(s) I'm not entirely sure how to go about proving this, though. What I did was what my professor called "preliminary analysis" -- evaluating an adequate delta to use to do the actual proof. The actual proof...the part I don't know how to do My work shown is just the profession of the algebra needed to find a delta which is used in the proof. I'm basically following my textbook's example here. I can't make sense of what my book is telling me to do past this point. __________________ C://dos C://dos.run run.dos.run
 01-26-09, 09:55 PM #5 jschen riding once again     Join Date: Oct 2005 Location: San Diego, CA Bikes: '06 Cervelo R3, '05 Specialized Allez Posts: 7,359 Mentioned: 0 Post(s) Tagged: 0 Thread(s) Quoted: 0 Post(s) Okay, let's define the parameters of the proof. To show that lim (x --> a) f(x) = y, you need to demonstrate that for any arbitrarily small deviation from the output value e (we don't know anything about e other than that it's arbitrarily small, which helps cancel out some terms), we can find a value d such that for all input values that differ from a by no more than d, the output deviates from y by no more than e. Reword the highly precise (but rather arcane) sentence in your mind (or on paper) in plain English. Then write the same idea in mathematical symbols. Or, just Google "limit proof" (without the quotes) and go at it without understanding the concept. If you choose the former, let me know what you come up with. If the latter, well, good luck. __________________ If you notice this notice then you will notice that this notice is not worth noticing.
 01-26-09, 09:59 PM #6 jschen riding once again     Join Date: Oct 2005 Location: San Diego, CA Bikes: '06 Cervelo R3, '05 Specialized Allez Posts: 7,359 Mentioned: 0 Post(s) Tagged: 0 Thread(s) Quoted: 0 Post(s) Oh, one more hint. Since e can be arbitrarily small, d almost certainly (though not always) will have some dependence on e. (In a trivial case, to prove that lim (x --> 3) 0 = 0, you can use whatever d you want since 0 = 0 for all values of x, so there is no need for d to have any dependence on e.) Okay, go at it. I'm not saying anything more until you come up with something, ANYTHING, that helps you out a bit (no matter how little it helps). __________________ If you notice this notice then you will notice that this notice is not worth noticing.
 01-26-09, 10:00 PM #7 x136  phony collective progress     Join Date: Sep 2006 Location: San Hoosey Bikes: http://velospace.org/user/36663 Posts: 2,981 Mentioned: 0 Post(s) Tagged: 0 Thread(s) Quoted: 0 Post(s) Do all the math you want, but the answer will still be 42. __________________