<s>[INST] Evaluating the limit $\lim_{x\to1}\left(\frac{1}{1-x}-\frac{3}{1-x^3}\right)$ In trying to evaluate the following limit:
$$\lim_{x\to1}\left(\frac{1}{1-x}-\frac{3}{1-x^3}\right)$$
I am getting the indefinite form of:
$$\frac{1}{\mbox{undefined}}-\frac{3}{\mbox{undefined}}$$
What would be the best solution to evaluating this limit?
 [/INST] Hint. Note that for $x\not=1$, 
$$\frac{1}{1-x}-\frac{3}{1-x^3}=\frac{x^2+x+1-3}{(1-x)(x^2+x+1)}
=\frac{(x+2)(x-1)}{(1-x)(x^2+x+1)}=-\frac{x+2}{x^2+x+1}.
$$
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