<s>[INST] limit with $\arctan$ I have to find the limit 
and want ask about a hint:
$$\lim_{n \to \infty} n^{\frac{3}{2}}[\arctan((n+1)^{\frac{1}{2}})- \arctan(n^{\frac{1}{2}})]$$
I dont have idea what to do. Derivatives and L'Hôpital's rule are so hard
 [/INST] Here is another approach, just compute the Taylor series of $\arctan((x+1)^{\frac{1}{2}})- \arctan((x)^{\frac{1}{2}}$ at the point $x=\infty$ 
$$ \arctan((x+1)^{\frac{1}{2}})- \arctan((x)^{\frac{1}{2}}= \frac{1}{2}\, \frac{1}{x^{3/2}}-\frac{7}{8}\, \frac{1}{x^{5/2}}+O \left(  \frac{1}{x^{7/2}} \right). $$
So the above expansion gives
$$ n^{3/2}(\arctan((n+1)^{\frac{1}{2}}) - \arctan((n)^{\frac{1}{2}})= \frac{1}{2}\, -\frac{7}{8}\, \frac{1}{n}+O \left(  \frac{1}{n^{2}} \right)$$
$$ \implies \lim_{n\to \infty} n^{3/2}( \arctan((x+1)^{\frac{1}{2}})- \arctan((x)^{\frac{1}{2}})=\frac{1}{2}. $$ 
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