Answer
$\dfrac{28}{11}$
Work Step by Step
Recall the limit rules:
$ \ Rule -\ 1 \ : \lim\limits_{x \to a} [A(x)]^n =[\lim\limits_{x \to a} A(x)]^n $
$\ Rule \ - \ 2 : \lim\limits_{x \to a} A(x)=A(a)$
$\ Rule \ - \ 3: \lim\limits_{x \to a} \dfrac{A(x)}{B(x)}=\dfrac{\lim\limits_{x \to a} A(x)}{\lim\limits_{x \to a} B(x)}$
$\lim\limits_{x\to 3}\dfrac{x^4-3x^3+x-3}{x^3-3x^2+2x-6}\\=\lim\limits_{x\to 3} \dfrac{x^3(x-3)+1(x-3)}{x^2(x-3)+2(x-3)}$
Apply $\lim\limits_{x \to a} \dfrac{A(x)}{B(x)}=\dfrac{\lim\limits_{x \to a} A(x)}{\lim\limits_{x \to a} B(x)}$
$\lim\limits_{x\to 3}\dfrac{(x-3)(x^3+1)}{(x-3)(x^2+2)}=\lim\limits_{x\to 3}\dfrac{(x^3+1)}{(x^2+2)}\\=\dfrac{27+1}{9+2} \\=\dfrac{28}{11}$
