Is this what you mean?
 \lim_{x \to \01}  \frac{2 x^{4}-2 x^{3} - x^{2} +1}{ x^{4} - x^{2} -2x+2}

By factoring the two polynomials, we get
 \lim_{x \to \01}  \frac{(1-2x+ x^{2} )(1+2x+2 x^{2} )}{(1-2x+ x^{2} )(2+2x+x^{2} )}

Canceling, this will remain:
 \lim_{x \to \01}  \frac{(1+2x+2 x^{2} )}{(2+2x+x^{2} )}

Then, we take the limit of the numerator. And divide it with the limit of the denominator.

 \frac{\lim_{x \to \01} 1+2x+2 x^{2} }{\lim_{x \to \01} 2+2x+x^{2} }

Since the function is continuous, this counts as just substituting 1 to the values of x. 

We get:
= \frac{5}{5}

Therefore, the limit is 1.