<s>[INST] Find the limit of function using Taylor series Good evening, 
I'm somehow stuck on solving some easy exercises :
$$\lim_{x\to\infty} x^{3/2}\bigl(\sqrt{x+1}+\sqrt{x-1}-2\,\sqrt{x}\bigr)$$
 [/INST] $$\sqrt{x+1}-\sqrt{x}=\frac{1}{\sqrt{x}+\sqrt{x+1}},\qquad \sqrt{x-1}-\sqrt{x}=-\frac{1}{\sqrt{x}+\sqrt{x-1}},$$
hence:
$$\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}=\frac{\sqrt{x-1}-\sqrt{x+1}}{(\sqrt{x}+\sqrt{x+1})(\sqrt{x}+\sqrt{x-1})}=\frac{-2}{(\sqrt{x}+\sqrt{x+1})(\sqrt{x}+\sqrt{x-1})(\sqrt{x-1}+\sqrt{x+1})}\approx\frac{-2}{(2\sqrt{x})^3}$$
and the limit is $\color{red}{-\frac{1}{4}}$.
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