Answer
$\dfrac{2}{3}$.
Work Step by Step
As we can see from the attached graph, as $x$ gets closer and closer to $1$ from the left and the right, the value of the function (the y-value) gets closer and closer to $\frac{2}{3}$. Thus the limit is
$\displaystyle \lim_{x\to1}\frac{x^3-x^2+x-1}{x^4-x^3+2x-2}=\frac{2}{3}$
