Prove that limx rightarrow 0 cos 1x doesnt existSolutionLet

Prove that lim_x rightarrow 0 cos 1/x doesn\'t exist.

Solution

Let us prove by contradictory method:

Let us assume that the limit exist

Therefore let us consider By the definition of limit, for this limit to exist there should exist a >0 for any >0 such that:

|cos1xL |<

whenever,

|x|<

When x takes small values cos1/x fluctuates rapidly between 1 and 1. So if we fix an arbitrarily small value for , we can always choose an x such that

|x|<

But

|cos1xL|>

which is a contradiction

So the limit does not exist.