Ok, let's assume we have a continuous function f:[0,infty) --> R such that lim_{x --> infty} f(x) = A, for some finite value A. (I have shown that A is 1 for our example, and that x*sin(1/x) (0 in 0) is continuous as well. These are the only properties I will need to show it uniformly continuous.)

The basic idea is to use that f is already uniformly continuous on each compact interval [0,N] and that (for large N) f(x) is almost equal to A, so that nothing "bad" can happen there: if all values are close to A, then they are certainly close to each other.

So take e>0. I will need to find a d>0 such that when |x-x'| < d then |f(x) - f(x')| < e (regardless which x and x' from [0,infty) I choose).

So take N so large that |f(x) - A| < e/2. This N exists (by definition) because of the limit of f at infinity being A. Now consider the interval I=[0,N+1]. The function f is uniformly continuous here, so that there exists a d>0 such that (x and x' from I) and |x-x'| < d imply that |f(x)-f(x')| < e. We assume without loss of generality that d < 1 (we can always choose a smaller one)

Now look at any two values x and x' from [0,infty) with |x-x'| < d. If both are in I, we are ok. The fact that d < 1 implies that if x and x' are d-close together, then they are either both in I or both >=N: the overlap of [N,infty) and [0,N+1] is too big. (draw a picture). And if they are both >= N then |f(x)-f(x)| <= |f(x)-A| + |A-f(x')| < e/2 + e/2 = e. So in either case, |f(x)-f(x')| < e and we are done: d works.

Ask-a-doc Web sites: If you've got a quick question, you can try to get an answer from sites that say they have various specialists on hand to give quick answers... Justanswer.com.

JustAnswer.com...has seen a spike since October in legal questions from readers about layoffs, unemployment and severance.

Web sites like justanswer.com/legal ...leave nothing to chance.

Traffic on JustAnswer rose 14 percent...and had nearly 400,000 page views in 30 days...inquiries related to stress, high blood pressure, drinking and heart pain jumped 33 percent.

Tory Johnson, GMA Workplace Contributor, discusses work-from-home jobs, such as JustAnswer in which verified Experts answer people’s questions.

I will tell you that...the things you have to go through to be an Expert are quite rigorous.

What Customers are Saying:

My Expert answered my question promptly and he resolved the issue totally. This is a great service. I am so glad I found it I will definitely use the service again if needed.One Happy CustomerNew York

My Expert answered my question promptly and he resolved the issue totally. This is a great service. I am so glad I found it I will definitely use the service again if needed.One Happy CustomerNew York

I am very happy with my very fast response. Eric is very knowledgeable in the subject area. Thank you! RPAustin, TX

Hi John, Thank you for your expertise and, more important, for your kindness because they make me, almost, look forward to my next computer problem. After the next problem comes, I'll be delighted to correspond again with you. I'm told that I excel at programing. But system administration has never been one of my talents. So it's great to have an expert to rely on when the computer decides to stump me. God bless, BillBill M.Schenectady, New York

The Expert answered my Mac question and was patient. He answered in a thorough and timely manner, keeping the response on a level that could understand. Thank you! FrankCanada

Wonderful service, prompt, efficient, and accurate. Couldn't have asked for more. I cannot thank you enough for your help. Mary C.Freshfield, Liverpool, UK

This expert is wonderful. They truly know what they are talking about, and they actually care about you. They really helped put my nerves at ease. Thank you so much!!!!AlexLos Angeles, CA

Thank you for all your help. It is nice to know that this service is here for people like myself, who need answers fast and are not sure who to consult.GPHesperia, CA

Meet The Experts:

Andy

Computer Consultant

Satisfied Customers:

5316

11yr exp, Comp Engg, Internet expert, Web developer, SEO