This winds up being an application of something called the "squeeze theorem." You take two expressions, one that is less than the expression you want the limit for and one that is greater. Then if, as you apply your limit, the limit of the lesser and greater expressions are the same then the limit of your expression, the one in the middle, must be the same as these.
So 1 is always less than
is always less than
. As you take the limit as
we find that
so
What your instructor is calling the "factor formula" I call by the name of a "sum to product" formula. By using the sum of angles formulas for sine and cosine
we may construct the following formulas:
For
we may use the top expression with
and
:
-Dan