Derivative of monotone increasing and bounded f

[losin]

Let f is monotone increasing, bounded, and differentiable on (a,inf)

Then does it necessarily follow that lim(f'(x),x,inf)=0 ?

It is hard to guess intuitively or imagine a counterexample...

 

[sutupidmath]

I will give you an 'intuitive' hint of what is going on. Since the function f is increasing and most importantly bounded, it means that as x-->infty, f(x) aproaches some point. IN other words |f(x)|<M, for some M, for all x. When you are taking the derivative of f, you are really talking about the slope of the tangent line at each point of f. since f is bounded by M, it means that as x->infty, f must get flater and flater. as a result the slopes of the tangent lines must also get smaller and smaller, eventually approaching zero.

In this case if A={f(x)|x in (a,infty)}, then f(x)-->sup{A} as x-->infty.