Answer to Question #18352 in Real Analysis for Matthew Lind
Using the definition of the limit at infinity verify that
0 does not equal lim x-->infinity cosx.
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.