<s>[INST] Find $\lim\limits_{x \to \infty} 2x(\sqrt{x-1} - \sqrt{x+5})$ $\lim\limits_{x \to \infty} 2x(\sqrt{x-1} - \sqrt{x+5})$
For what i've found the part in brackets is an indeterminate form. I've tried to multiply the bracket part by $\frac{\sqrt{x - 1} + \sqrt{x+5}}{\sqrt{x - 1} + \sqrt{x+5}}$ , then multiply the numerator by $2x$. I don't know what to do next.
 [/INST] You follow the right approach.
\begin{align}
f(x) &= 2x (\sqrt{x-1}-\sqrt{x+5}) = 2x 
\frac{(\sqrt{x-1}-\sqrt{x+5})\cdot (\sqrt{x-1}+\sqrt{x+5})}{(\sqrt{x-1}+\sqrt{x+5})}
\\
&=2x\frac{x-1-x-5}{(\sqrt{x-1}+\sqrt{x+5})} = 2x\frac{-6}{(\sqrt{x-1}+\sqrt{x+5})}
=  -12\frac{x}{(\sqrt{x-1}+\sqrt{x+5})}
\end{align}
Now you can try to bound $f$ from below and above by an expression of the form $$c\frac{x}{\sqrt{x}}$$
What do you know about an expression of that form?
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