<s>[INST] Limit of $\left(y+\frac23\right)\mathrm{ln}\left(\frac{\sqrt{1+y}+1}{\sqrt{1+y}-1} \right) - 2 \sqrt{1+y}$ We consider the limit for large $y$ of the following expression :
$$\left(y+\frac23\right)\mathrm{ln}\left(\dfrac{\sqrt{1+y}+1}{\sqrt{1+y}-1} \right) - 2 \sqrt{1+y}.$$
Many references state that the large $y$ behaviour of this last expression is $y^{-3/2}$. I have been trying to show this in any way possible but I'm only hitting dead ends. Wolframalpha gives a Puiseux series of that expression which makes this behaviour explicit, but I do not understand how to get to this series either. Could you please help me on this?
 [/INST] Set $x=\sqrt{1+y}-1$. Then $y=(x+1)^2-1=x^2+2x$, and your expression simplifies to
$$ \bigl(x^2+2x+\tfrac23 \bigr)\log\bigl(1+\tfrac2x \bigr) - 2x - 2 $$
Expanding the logarithm in powers of $2/x$ gives us
$$ \bigl(x^2+2x+\tfrac23 \bigr)\Bigl(\frac2x-\frac2{x^2}+\frac{8}{3x^3}-\frac{4}{x^4}+\frac{32}{5x^5}+o(x^{-5}) \Bigr) - 2x - 2 $$
We can then multiply out and collect terms to get
$$ \begin{align} (2-2)&x^1 + {}
\\ (-2 + 4 - 2)&x^0 +{}
\\ (\tfrac83 -4 + \tfrac43)&x^{-1} +{}
\\ (-4+\tfrac{16}3 - \tfrac43)&x^{-2} +{}
\\ (\tfrac{32}5 -8 + \tfrac{16}{9})&x^{-3} + o(x^{-3})
= \tfrac{8}{45}x^{-3} + o(x^{-3})
\end{align}
$$
Since asymptotically $x\sim y^{1/2}$ this is also
$$\frac{8}{45}y^{-3/2} + o(y^{-3/2}).$$
</s>