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[tex] \lim_{x \to \01} \frac{2 x^{4}-2 x^{3} - x^{2} +1}{ x^{4} - x^{2} -2x+2} [/tex]
By factoring the two polynomials, we get
[tex] \lim_{x \to \01} \frac{(1-2x+ x^{2} )(1+2x+2 x^{2} )}{(1-2x+ x^{2} )(2+2x+x^{2} )} [/tex]
Canceling, this will remain:
[tex] \lim_{x \to \01} \frac{(1+2x+2 x^{2} )}{(2+2x+x^{2} )} [/tex]
Then, we take the limit of the numerator. And divide it with the limit of the denominator.
[tex] \frac{\lim_{x \to \01} 1+2x+2 x^{2} }{\lim_{x \to \01} 2+2x+x^{2} } [/tex]
Since the function is continuous, this counts as just substituting 1 to the values of x.
We get:
=[tex] \frac{5}{5} [/tex]
Therefore, the limit is 1.
