Answer
$\dfrac{8}{9}$
Work Step by Step
Recall that if $ f $ is a polynomial function, then we have $\lim_\limits{x\to a}f(x)=f(a)$.
In order to to find the limit, we will plug $ a $ into the function and then simplify.
$\lim_\limits{x\to 2} \dfrac{x^3-2x^2+4x-8}{x^4-2x^3+x-2}=\dfrac{\lim_\limits{x\to 2}
(x^2+4) (x-1)}{\lim_\limits{x\to 2} (x-2)(x+1)(x^2-x+1)}=\lim_\limits{x\to 2} \dfrac{x^2+4}{(x+1)(x^2-x+1)}=\dfrac{8}{3 \cdot 3}=\dfrac{8}{9}$
