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While generally analyze the functional series so there is the ultimate techniques how to deal with them. There is square root criterion, unit test, comparison with $ \frac{1}{a_n ^ q} $ and limit their options. I tried to find their equivalent in sequence but I have not found anything so your ultimate.

Is there some common almost always working techniques how to calculate the limit of function sequence?

For example 1: Find the limit of $arctan(x+n)$

The only way how I can do it is to draw first few elements of this Wolfram calculate for example 1 and I will se that this sequence will converge to $\pi/2$. That is not much exact solution. Just look and see.

For example 2: Find the limit of $arctan(x/n)$

I will draw first few elements of this series Wolfram calculate for example 2 and I see it will converge to $0$.

For example 3: Find the limit of $arctan(x*n)$

I will draw first few elements of this series Wolfram calculate for example 3 Wolfram calculate for example 3 and I see it will converge to $\frac{\pi}{2}sign(x)$.

You can see that in all examples nothing was proved. It is just look and see ...

Is there some better (exact) ways how to deal with function sequences in general? Please show it on examples 1, 2, 3.

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Instead of looking at a screen, one can look inward and see. Formal details depend on the exact definition used for $\arctan$. For $\arctan(xn)$ the formula in the post needs to be modified. –  André Nicolas Mar 7 '13 at 8:51

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