The real limit of a function #f(x)#, **if it exists**, as #x->oo# is reached no matter how #x# increases to #oo#. For instance, no matter how #x# is increasing, the function #f(x)=1/x# tends to zero.

This is not the case with #f(x)=cos(x)#.

Let #x# increases to #oo# in one way: #x_N=2piN# and integer #N# increases to #oo#. For any #x_N# in this sequence #cos(x_N)=1#.

Let #x# increases to #oo# in another way: #x_N=pi/2+2piN# and integer #N# increases to #oo#. For any #x_N# in this sequence #cos(x_N)=0#.

So, the first sequence of values of #cos(x_N)# equals to #1# and the limit must be #1#. But the second sequence of values of #cos(x_N)# equals to #0#, so the limit must be #0#.

But the limit cannot be simultaneously equal to two distinct numbers. Therefore, there is no limit.