<s>[INST] Evaluate $\lim\limits_{x \to 1} \left( \frac{1}{x-1}-\frac{3}{1-x^3} \right)$ Evaluate $$\lim_{x \to 1} \left( \frac{1}{x-1}-\frac{3}{1-x^3} \right)$$
My attempt: $$\lim_{x \to 1} \left( \frac{1}{x-1}-\frac{3}{1-x^3} \right) = \lim_{x \to 1} \frac{x+2}{x^2+x+1}=1$$
According to the answer key, this limit does not exist. I turned that into one fraction, then I factored the polynomial on the numerator as $-(x-1)(x+2)$ and the one on the denominator as $(x+1)(-x^2-x-1)$. What did I do wrong?
 [/INST] $$\frac1{x-1}-\frac3{1-x^3}=\frac1{x-1}+\frac3{x^3-1}\to\infty$$
as the terms do not cancel each other (they have the same sign).

Assuming a typo,
$$\frac1{x-1}+\frac3{1-x^3}=\frac1{x-1}-\frac3{x^3-1}=\frac{x^2+x-2}{x^3-1}=\frac{x+2}{x^2+x+1}\to1.$$
</s>