for the problem above, is the idea that u can multiply as many times by 1 as you need to to get the form sinx/x or x/sinx?
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Basically, yes. Until you learn L'Hôpital's rule, this is the way you must handle these types of problems, by algebraic manipulation.
The idea is that yes you multiply by 1, but remember that t is approaching zero which means that the limit has to be equal to 1 for these terms and in case it is.
If these new limits were not equal to 1 then you couldn't do the substitution.
Yes that is exactly the idea. Though I would simply do this...
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