<s>[INST] Iterated limits of $\frac{x-y}{x^3-y}$ Why it the following limits look like this: $$\lim_{x\rightarrow -1} \frac{x-y}{x^3-y}=1$$ but suprisingly  $$\lim_{y\rightarrow -1} \frac{x-y}{x^3-y}=\frac{1}{1-x+x^2}$$I thought that after substituting the value of $y=-1$ the limit will be equal $\frac{1}{x^2}$. I do not know what am i doing wrong.
 [/INST] $$\lim_{x\to -1} \frac{x-y}{x^3-y}=\frac{-1-y}{-1-y}=1$$
Whereas for your second limit,
$$\lim_{y\to -1} \frac{x-y}{x^3-y}=\frac{x+1}{x^3+1}$$
Using the identity $a^3+b^3=(a+b)(a^2+b^2-ab)$,
$$\frac{x+1}{x^3+1}=\frac{1}{x^2-x+1}$$
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