Can Limits to Infinity Prove a Zero Derivative Over the Reals?

[JPBenowitz]

Suppose I have

lim_{t\rightarrow∞} f(g(t)) = 0

and

lim_{t\rightarrow-∞} f(g(t)) = 0

How would I prove \frac{df}{dt}_{|}\Re = 0? (over the reals)

 

[theorem4.5.9]

This question doesn't make sense. Could you provide some context?

 

[JPBenowitz]

This question doesn't make sense. Could you provide some context?

I'm having problems displaying what I want to convey. Basically I proved that the limit for the curvature function will always converge to zero for any real continuous function and now I want to prove that the derivative with respect to time will always converge to zero.

So, essentially I need to prove that the integral over the reals of the curvature function will always converge to some constant.

 

[algebrat]

deleted