f-0-1-R-lim-x-0-f-x-0-lim-x-0-f-x-f-x-2-x-0-lim-x-0-f-x-x-

Question Number 56706 by gunawan last updated on 22/Mar/19

f : [0, 1) \to R

\lim_{x \to 0} f (x) = 0

\lim_{x \to 0} \frac{f (x) - f (\frac{x}{2})}{x} = 0

\lim_{x \to 0} \frac{f (x)}{x} = \dots

Commented by Abdo msup. last updated on 22/Mar/19

\frac{x}{2} = t \Rightarrow {l i m}_{x \to 0} \frac{f (x) - f (\frac{x}{2})}{x} = {l i m}_{t \to 0} \frac{f (2 t) - f (t)}{2 t} = 0

\Rightarrow {l i m}_{t \to 0} \frac{f (2 t)}{2 t} = \frac{1}{2} {l i m}_{t \to 0} \frac{f (t)}{t} \Rightarrow l = \frac{l}{2} \Rightarrow l = 0 \Rightarrow

{l i m}_{x \to 0} \frac{f (x)}{x} = 0