Answer
$\dfrac{28}{11}$
Work Step by Step
Factor each polynomial:
\begin{align*}
\lim_{x\to 3}\frac{x^4-3x^3+x-3}{x^3-3x^2+2x-6}&=\lim_{x\to 3}\frac{x^3(x-3)+(x-3)}{(x-3)x^2+2(x-3)}\\&=\lim_{x\to 3}\frac{(x-3)(x^3+1)}{(x-3)(x^2+2)}.\end{align*}
Cancel the common factors, then substitute $3$ to $x$:
\begin{align*}
\lim_{x\to 3}\frac{x^3+1}{x^2+2}&=\frac{3^3+1}{3^2+2}\\&=\frac{28}{11}
\end{align*}
