# Answer to Question #18352 in Real Analysis for Matthew Lind

Question #18352

Using the definition of the limit at infinity verify that

0 does not equal lim x-->infinity cosx.

0 does not equal lim x-->infinity cosx.

Expert's answer

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.

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.

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