Understand Decreasing Function: Limit g'(x) at Infinity

[Sethka]

Now what should I look up to understand a question like this one:

Function h is decreasing on the interval [4,infinity) and lim h(x)(with x approaching infinity)=8, what would be the limit g'(x)(with x approaching infinity)?

I don't nessicarily need the answer, but could someone point me in the right direction?

 

[HallsofIvy]

Earlier you asked about a function g that was simply decreasing and were told that you cannot say anything about the derivative as x went to infinity except that it would be non-negative (the limit could be 0).

Is this a revision of that problem? If so, it is a good revision. (I won't make any remark about not being able to find g' from information about h!) The difference now is that h goes to a constant as x goes to infinity- that's more important than "decreasing". Given any \epsilon> 0, there must exist a "neighborhood of infinity" (interval (a, \infty)) on which h(x) differs less than \epsilon from 8. What does that tell you about the derivative of h?