As .
The cosine function oscillates between , so there is not a limiting value.
I don't understand what you mean by the bolded line, but your argument can not work - you have that , but if x is positive then this means that a positive quantity (LHS) is smaller than a negative quantity (RHS), which is absurd.
This is an outline of the proof I have in mind:
1) Suppose towards a contradiction that the limit exists. Then since cosine is bounded, the limit is finite, say L.
2) Find two sequences such that as , and such that and , and argue why this finishes the proof.
Edit: Notice that 2) was wrong at first, fixed it now.
"Let cosine be sufficiently large"??
Proof by contradiction. Suppose the limit does exist. Let . Then, given any , there exist R such that if x> R, . Certainly, we must have .
Case 1: . Let . There exist an integer n such that . For , so [tex]|cos(x)- L|= L+1> 1- L, a contradiction.
Case 2: . Let . Then there exist an integer n such that . You finish.