L'Hopital:
The radical term can be eliminated because it evaluates to 1:
The cos term in the denominator evaluates to 1 and we have:
So, we have
Perhaps there's a mistake since the limit, as given, doesn't exist: the left and right limits are different, but if we look for the limit when then:
your idea looks just fine to me, but instead of writing the MacClaurin series for sine and cosine let us use the series for , and now we can try L'Hospitals rule on that fraction, since , so derivating numerator and denominator we get:
, and thus the limit is
Tonio