calculus and analysis - Difficulty with computing a limit

I meet some difficulties when trying to compute this limit,

Limit[1/n*Integrate[1/(Cos[x]^2 + 4 Sin[2 x] + 4), {x, 0, Pi*n}],   n -> Infinity,Assumptions -> n \[Element] Integers]

The output is,

Limit::cas: Warning: Contradictory assumption(s) (n\[Element]Integers&&(Re[n]<=1/2||n\[NotElement]Reals))&&n>4096 encountered. >>

The limit is finite though.

Answer

Break it up:

int = Assuming[Element[n, Integers],
          Integrate[1/(Cos[x]^2 + 4 Sin[2 x] + 4), {x, 0, Pi*n}]]
(*  (n*Pi)/2  *)

now

Limit[1/n*int, n -> Infinity]
(* Pi/2 *)

You do not even need limit.

(1/n)*int

(*  Pi/2 *)

To do it as you did, you need to put the Assuming first, so it covers the integral part, like this

Assuming[Element[n, Integers], 
 Limit[1/n*Integrate[1/(Cos[x]^2 + 4 Sin[2 x] + 4), {x, 0, Pi*n}],n -> Infinity]]

(* Pi/2  *)

What you had is this:

Limit[1/n*Integrate[1/(Cos[x]^2 + 4 Sin[2 x] + 4), {x, 0, Pi*n}],   
      n -> Infinity,Assumptions -> n \[Element] Integers]

So, the Integrate part never knew that n was an integer ! This is important. Since without this information, integrate will generate an answer this like this:

int = Integrate[1/(Cos[x]^2 + 4 Sin[2 x] + 4), {x, 0, Pi*n}]
(* 1/2 (-ArcTan[2] + ArcTan[2 (1 + Tan[n Pi])]) *)

Assuming[Element[n, Integers], Limit[1/n*int, n -> Infinity]]

(* 0 *)

Compare the result on Integrate when it sees the assumption on n being integer:

Assuming[Element[n, Integers], Integrate[1/(Cos[x]^2 + 4 Sin[2 x] + 4), {x, 0, Pi*n}]]
(* (n Pi)/2  *)

big difference

1/2 (-ArcTan[2] + ArcTan[2 (1 + Tan[n Pi])])  

vs

(n Pi)/2