<s>[INST] How can find this limit? $\lim_{x\to-∞}(\sqrt{x^2+6x}-\sqrt{x^2-2x})$ $$\lim_{x\to-∞}(\sqrt{x^2+6x}-\sqrt{x^2-2x})$$
plugging in infinity gives infinity - infinity, what kind of manipulation can I do to solve this?
 [/INST] Note that $$ \sqrt{x^2+6x}-\sqrt{x^2-2x} = \frac{8x}{\sqrt{x^2+6x}+\sqrt{x^2-2x}}, $$so our answer is the limit of the above as $ x \to -\infty $. But note that $ \sqrt {x^2 + 6x} + \sqrt {x^2 - 2x} \sim \mathcal{O}(2x) $, so the answer is $\frac{8}{2}=\boxed{4}$. 
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