How do you determine the limit of #cos(x) # as n approaches #oo#?

1 Answer
Apr 6, 2016

There is no limit (the limit does not exist).

Explanation:

As #x# increases without bound it takes on values of the form #pik# infinitely many times for even integer #k# and infinitely many times for odd integer #k#.

Therefore, as #x# increases without bound, #cosx# takes on values of #1# and of #-1#, infinitely many times.

It is not possible that there is an #L# that #cosx# is within #epsilon = 1/2# of for all #x# greater than some specified #N#.
Less formally, there is no #L# that all of the values of #cosx# are eventually close to.
Even less formally, the values of #cosx# cannot zero in on a particular #L# while also going all the way from #-1# to #1# and back again.

Finally, here is the graph:

graph{cosx [-9.75, 82.75, -22.43, 23.84]}