Question

Using the definition of the limit at infinity verify that
0 does not equal lim x-->infinity cosx.

Answer 1

The assumption
limx-->infinity cosx = 0
means that for any epsilon>0 there exists N>0 suchthat for all x>N we have that
|cos x| < epsilon.
Consider the seuqence of numbers x_n = 2 pi n.
Then
lim x_n = +infty
and
cos x_n = 1 for all n.
Take epsilon < 1.Then for any N>0 there exists n such that
x_n = 2 pi n >N
and
cos x_n = 1 >epsilon.
Therefore 0 does not equal lim x-->infinity cosx.