Answer
True.
Work Step by Step
It is given that, if $\lim\limits_{x \to 1} \frac{f(x)-f(1)}{x-1}=3$.
Step-1: Let $x=h+1 \implies h=x-1$. Thus, $x \to 1$, $h \to 0$.
Step-2: Therefore, our limit becomes $$\lim\limits_{h \to 0} \frac{f(h+1)-f(1)}{h}=3$$
