<s>[INST] Compute $\lim\limits_{x\to \infty} \left(\sqrt{\left(x+\frac{1}{\sqrt x}\right) ^3} - \sqrt{x^3} \right) $ Compute $\lim\limits_{x\to \infty} \left(\sqrt{\left(x+\frac{1}{\sqrt x}\right) ^3} - \sqrt{x^3} \right) $.
I tried multiplying by the conjugate and I get to an ugly expression. What should I do? 
 [/INST] So, you arrived at
$$\begin{align}
\sqrt{(x+1/\sqrt x)^3}-\sqrt{x^3}&=\frac{3x^{3/2}+3+x^{-3/2}}{\sqrt{(x+1/\sqrt x)^3}+\sqrt{x^3}}\\\\
&=\frac{3x^{3/2}+3+x^{-3/2}}{x^{3/2}\left(1+\sqrt{(1+1/x^{3/2})^3}\right)}
\end{align}$$
Now let $x\to \infty$.  Can you finish now?
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