t = h/2
h-> 0 => t -> 0

lim[h->0] 1/h{f(x+h/2)-f(x)}
= lim[t->0] 1/2t{f(x+t)-f(x)} [h=>t]
= 1/2 lim[t->0] 1/t{f(x+t)-f(x)}
= 1/2f'(x)
こうですか？