Hello, I have been looking around the net recently trying to teach myself how to do delta/epsilon proofs. unfortunately, my prof sucks . Anyway, I think I understand how to get delta in terms of epsilon for limits where x approaches a number, but I'm having difficulty finding what to do when x approaches a value 'a'. eg. prove: lim of x approaches a ( 1/x^3) = (1/a^3) I don't really need a solution, although it would be welcome. I'm just interested in what the usual procedure is for finding delta as I'm always stuck with an 'a' term and an 'x' term that I'm not sure what to do with or how to get rid of. Any help is greatly appreciated.
i usually write x as a+h, whgere h is basically delta, and then subtract f(x) from f(a) to see how big it is. then i try to make it elss than epsilon. for example if i have f(x) = 1/x, i call it 1/(a+h) and look at 1/(a+h) - 1/a. assume a >0. that equals (a - [a+h])/[a(a+h)] = -h/[a(a+h)]. now i want this to be small, so i want the bottom to be large, and i need to get a handle on h. If h were too close to -a the bottom would get too small so I choose h first of all to be less than a/2. Then a+h is between a/2 and 3a/2. so the bottom is between a^2/2 and 3a^2/2. remember i want the bottom to be large, so i know the bottom is at least as large as a^2/2, whenever h is less than a/2. now that emans the fraction h/[a(a+h)], is at least as small as h/[a^2/2] = 2h/a^2. so if i can also make 2h/a^2 less than epsilon i am home. well that is easy to solve for. i need h less than a^2/2 times epsilon, and also less than a/2. i hated these fraction ones as a young student because you have to just play around with them likwe this, but see if you can check this out and then if it works, generalize it to your case.
Saying your prof sucks isn't going to get any sympathy since most of the people here who answer questions are teachers who probablyu doubt your ability to make an unbiased judgement about someone trying to teach you a difficult subject that you don't yet understand. As for this question, or many analysis questions, finding the exact eps-delta stuff is completely unnecessary: you know x tends to a as, erm, x tends to a, and that if x and y tend to a and b resp that xy tends to ab, and that if a is not zero 1/x tends to 1/a, hence result.
Do you have Thomas and Finney? It has a nice introduction to epsilon-delta and the formal definition of limit. You should be able to get a good deal of practice from that book as well. Cheers Vivek