Condition of the problem

Calculate the limit of the function:

$\lim _{x \rightarrow-1} \frac{\left(x^{3}-2 x-1\right)^{2}}{x^{4}+2 x+1}$
Solution

$\lim _{x \rightarrow-1} \frac{\left(x^{3}-2 x-1\right)^{2}}{x^{4}+2 x+1}=\left\{\frac{0}{0}\right\}=\lim _{x \rightarrow-1} \frac{\left(x^{2}-x-1\right)^{2}(x+1)^{2}}{\left(x^{3}-x^{2}+x+1\right)(x+1)}=$
$=\lim _{x \rightarrow-1} \frac{\left(x^{2}-x-1\right)^{2}(x+1)}{x^{3}-x^{2}+x+1}=\frac{\left((-1)^{2}-(-1)-1\right)^{2}(-1+1)}{(-1)^{3}-(-1)^{2}+(-1)+1}=$
$=\frac{(1+1-1)^{2} \cdot 0}{-1-1-1+1}=\frac{0}{-2}=0$

## Problem Kuznetsov Limits 10-6