$\displaystyle f $ is a differentiable function at $\displaystyle x=1$.prove that if $\displaystyle \lim_{h\rightarrow0}\frac{f(1+h)}{h}=1 $
so, $\displaystyle f(1)=0 $ & $\displaystyle f'(1)=1 $
You are given that both
$\displaystyle {\lim _{h \to 0}}\frac{{f(1 + h)}}{h}\quad \& \quad {\lim _{h \to 0}}\frac{{f(1 + h) - f(1)}}{h}$ EXIST.
Now doesn't that imply that $\displaystyle {\lim _{h \to 0}}\frac{{f(1)}}{h}$ must also exists?
If so, what does that tell you about $\displaystyle f(1)~?$ And why?