<s>[INST] How to solve indeterminations of the type $0/0$ I am unable to find these limits:
1)
$$
\lim_{x \to 1} \frac{3(1 - x^2) - 2(1 - x^3)}{(1 - x^3)(1 - x^2)}
$$
2)
$$
\lim_{x \to 0} \frac{\sqrt{1 - 2x - x^2} - (x + 1)}{x}
$$
3)
$$
\lim_{x \to 0} \frac{\sqrt{x + 2} + \sqrt{x + 6} - \sqrt{6} - \sqrt{2}}{x}
$$
4)
$$
\lim_{x \to 0} \frac{1 - \sqrt[3]{1 - x}}{1 + \sqrt[3]{3x - 1}}
$$
My interest is not in the answers, but in the algebraic manipulations i can use to eliminate the indeterminations of the type $0/0$.
My english skills are not so good, i'm sorry for this.
 [/INST] 1) Factorise both the numerator and denominator so that the terms $(1-x)^n$ can be simplified 
2) Use $$\alpha-\beta=\frac{\alpha^2-\beta^2}{\alpha+\beta}$$
3) Write
$$
\sqrt{x + 2} + \sqrt{x + 6} - \sqrt{6} - \sqrt{2}=(\sqrt{x + 2}  - \sqrt{2})+ (\sqrt{x + 6} - \sqrt{6})
$$
and use what we did in $2$.
4) Use $$\alpha-\beta=\frac{\alpha^3-\beta^3}{\alpha^2+\alpha\beta+\beta^2}$$
</s>