<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] Hello Taylor my old friend...
$\begin{array}\\
\sqrt{\left(x+\frac{1}{\sqrt x}\right) ^3} - \sqrt{x^3}
&=\left(x+x^{-1/2}\right) ^{3/2} - x^{3/2}\\
&=x^{3/2}\left(\left(1+x^{-3/2}\right) ^{3/2} - 1\right)\\
&=x^{3/2}\left(\left(1+\frac32x^{-3/2}+O(x^{-3})\right) - 1\right)\\
&=x^{3/2}\left(\frac32x^{-3/2}+O(x^{-3})\right)\\
&=\frac32+O(x^{-3/2})\\
\end{array}
$
Actually, of course,
it's the generalized binomial theorem,
but that doesn't scan as well.
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