Limits of sequences Find the limit of the following sequences or
determine that the limit does not exist.
1. {tan^-1n} SOMEONE HELP:(

Question

Limits of sequences Find the limit of the following sequences or 
determine that the limit does not exist.
1. {tan^-1n} SOMEONE HELP:(

anonymous · Answer

$ a_n = \tan^{-1}(n) ? $

buggiethebug · Answer

yes {tan^−1(n)}

buggiethebug · Answer

the answer is pi/2 and I don't see how.

anonymous · Answer

How did you introduce the tan function? The above function is the inverse of the tan function, that would be your answer if you understand about the domain and the codomain.

To make a function inversive it has to be bijective, meaning injective and surjective. the function $\tan x : \mathbb{R} \to \mathbb{R}$ is surjective, but not injective, how can we make it injective? We narrow down the domain to $ [-\pi /2, \pi/2 ]$.

anonymous · Answer

Thus the inverse function of tan called $\arctan$ or a bit vaguely written $\tan^{-1}$ maps from $\mathbb{R} \to [- \pi /2, \pi /2 ]$ by definition. So as the limit of n approaches infinity, $\arctan$ approaches $\pi/2$.

anonymous · Answer

Maybe this is not the exact answer you're looking for, however you should make sure that you understand that $\arctan$ is an 'artificial' function defined as the inverse function of $\tan$. If you know what it means to inverse a function (in plotting, uniqueness and so on) you will see that all results follow from that.

buggiethebug · Answer

I believe we were supposed to demonstrate it using The Squeeze Theorem, but this makes sense as well.