Epsilon - Delta Question

[blackhype]

I am familiar with solving epsilon delta proofs, but my professor handed me a problem and I have no idea what to do or where to begin. I guess its because I don't understand the wording of the problem or just the plain definition. I just need somebody to explain what exactly i'm supposed to do, maybe provide an example of your own.

Here is the problem:

1. If \(\displaystyle \bigl|\, x\, -\, 2\, \bigr|\, <\, \delta,\,\) show that the quantity \(\displaystyle \,\dfrac{(x\, -\, 2)\, x}{(x\, -\, 1)(x\, +\, 3)}\,\) can be bounded by \(\displaystyle \, k\delta\,\) for some constant \(\displaystyle \,k.\)

 

[daon2]

Please post only one question per thread. There is a problem with your first question. Obviously, the limit does not exist at x=2. Take \(\displaystyle \delta=1\) and let \(\displaystyle f(x)\) be the given rational function. Suppose such a k existed, then for all \(\displaystyle x\in(1,2)\cup(2,3), |f(x)|\le k\). But f has a vertical asymptote at x=2.

For the second question, it can be as simple as quoting continuity. Only you can tell us what was supposed to be done.

 

[blackhype]

I revised the problem, how do I go about solving it?

 

[daon2]

I revised the problem, how do I go about solving it?

You need to bound x so that it is not close to 1 or -3. Suppose \(\displaystyle \delta \le 1/2\). Then, use the assumption \(\displaystyle |x-2|<\delta\) to get lower and upper bounds for \(\displaystyle x, x-1, x+3\), and using these, finally find a bound for \(\displaystyle \left|\frac{x}{(x-1)(x+3)}\right|\), which will be your k.

One problem still does exist though. The statement is not true for all delta. If delta were at least 1, then you're again left with the problem of having a vertical asymptote.