The limit as x -> 0 of sin(tan(x)) / sin(x)
Is this supposed to be an intuitive question? Because I see no way I can manipulate this algebraically to get it into a useful form to cancel out anything. Otherwise, I don't see what the limit could be.
I keep forgetting to mention that this is a Calculus 1 course, therefore we are disallowed to apply rules we haven't learned yet. I should state that explicitly in my posts from now on, or at least put it in my signature.
I can only use algebra, and some special limits we learned (particularly sinx/x and x/sinx = 1 as x-> 0).
You don't need taylor series. tan(x) = sin(x)/cos(x), which is close to sin(x)/1, and its been shown that sin(x) behaves like x by formally stating that
I'm not saying everyone is expected to know this.. its just a useful strategy.