<s>[INST] Is this limit solution $-∞$? Is this correct?
$$\lim_{x\to\infty} x\left(\frac{\sqrt{x^2+6x+10}}{x^2}-1\right)=-∞$$
Please help me, I don't have the solution and I want to know if I solved it right.
 [/INST] I would write
$$\frac{1}{x}\left(\frac{\sqrt{x^2+6x+10}-x^2)(\sqrt{x^2+6x+10}+x^2)}{\sqrt{x^2+6x+10}+x^2}\right)$$
and you will get $$\frac{x\left(\frac{1}{x^2}+\frac{6}{x^3}+\frac{10}{x^4}-1\right)}{\sqrt{1+\frac{6}{x}+\frac{10}{x^2}}+1}$$.
This tends to $-\infty$ if $x$ tends to $\infty$
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