Show that $\displaystyle f(x)$ is the derivative of$\displaystyle f(.)$ at$\displaystyle x$ if and only if $\displaystyle lim_{h \to 0} \sup_{|t|\leqslant h} \frac{|f(x+t)-f(x)-tf^{'} (x)|}{h} = 0 $
I think I may have given you the wrong advice: I'm thinking that since h > 0 (limit approached from the right) then |h| = h so you can put the h inside the absolute value sign. I'm sorry about the wrong advice.
Now one approach I am thinking of is to find a situation when you can interchange the limit signs within the absolute value from outside. I do know there are theorems that gaurantee when you can do this in some situations like this:
http://www.math.cuhk.edu.hk/course/1...onvergence.pdf