<s>[INST] Evaluating $\lim_{x\to -\infty}\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2} \right )$ For some reason I fail to evaluate this (apparently) simple limit:
$$\lim_{x\to -\infty}\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2} \right )$$
I tried conjugate multiplication* however it didn't work for me. I thought about sandwiching, but I don't see how to do it here. I was also trying to evaluate it as a composition of functions but with no luck. Any suggestions?
*
Conjugate multiplication gives
$$\frac{2x}{\sqrt{1+x+x^2}+\sqrt{1-x+x^2}}$$
I tried working with this by factoring and canceling things out but it didn't work.
 [/INST] Conjugate multiplication will help and what you got is correct.
$$\lim_{x\to -\infty}\sqrt{1+x+x^2}-\sqrt{1-x+x^2}$$$$=\lim_{x\to -\infty}(\sqrt{1+x+x^2}-\sqrt{1-x+x^2})\cdot\frac{\sqrt{1+x+x^2}+\sqrt{1-x+x^2}}{\sqrt{1+x+x^2}+\sqrt{1-x+x^2}}$$
$$=\lim_{x\to -\infty}\frac{2x}{\sqrt{1+x+x^2}+\sqrt{1-x+x^2}}$$
(here setting $t=-x$ gives you)
$$=\lim_{t\to \color{red}{+}\infty}\frac{-2t}{\sqrt{1-t+t^2}+\sqrt{1+t+t^2}}$$
$$=\lim_{t\to\infty}\frac{-2}{\sqrt{\frac{1}{t^2}-\frac{1}{t}+1}+\sqrt{\frac{1}{t^2}+\frac 1t+1}}$$
$$=\frac{-2}{1+1}$$
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