<s>[INST] Solving a Limit Problem if the Limit Exists My question is, consider $\lim [(x^{2} +2x +1) / (x^{4} -1)]$ for $x \to -1$ (just arrow sign to the right)
What I have done so far is
$(x+1)^{2}/[(x^{2} +1)(x+1)(x-1)]$
$(x+1)/[(x^{2}+1)(x-1)]$
I'm not too sure what to do at this point. If I substitute in $x=-1$, I would get $0$.
So in this case would the limit not exist?
 [/INST] Note that $x^2+2x+1=(x+1)^2$ and $x^4-1=(x^2+1)(x+1)(x-1),$ 
so that $$\frac{x^2+2x+1}{(x^2+1)(x+1)(x-1)} = \frac{x+1}{x^2+1)(x-1)},$$ so $$\lim_{x \to -1} \frac{x^2+2x+1}{(x^2+1)(x+1)(x-1)} = \frac{0}{2 \cdot -2} = 0.$$
$0$ is a perfectly fine answer.  You may be confusing this with getting 0 in the denominator, which either means the limit doesn't exist or more work needs to be done.
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