$\underset{x\to 1}{lim} \frac{x^8-2x+1}{x^4-2x+1}$ equals

Solution:

We have, 

$\begin{array}{l}\underset{x\to 1}{lim} \frac{x^8-2x+1}{x^4-2x+1}\\ =\underset{x\to 1}{lim} \frac{(x-1)\left(x^7+x^6+x^5+x^4+x^3+x^2+x-1\right)}{(x-1)\left(x^3+x^2+x-1\right)}\\ =\underset{x\to 1}{lim} \frac{x^7+x^6+x^5+x^4+x^3+x^2+x-1}{x^3+x^2+x-1}=\frac{6}{2}=3\end{array}$