I came up with seven real ways of doing this limit...and one stupid way ...how many can you come up with? It's simple I know...but let see if we can get eight?
and as (*)
Multiply numerator and denominator by:
we have: and then apply (*)
3. Definition of where
( this is not valid in my opinion)
So if it's uniformly convergent (you'd have to go over that more carefully)
5. assuming it exists
You counted applying the Power Series expansion of cos(x), right?
Think of it this way. Suppose you were teaching a course in analysis. And you wanted to define the sine function. How would you define it? If you say "it is the ratio of the opposite to the the hypotenuse", that is no good. Because it is not formal enough. You need to define what it means "opposite" and "hypotenuse" but before doing that you need to define what it means "triangle" and before doing that you need to define what it means "length". All these words are not strict mathematical words. They are more geometric expressions. In high school you learn the sine function in terms of geometry but on a more formal level the sine function is just a power series.