[enriquto]

The limit in question 2 is wickedly difficult to solve by traditional means. You have to apply l'Hôpital rule about seven times, and it becomes a monster formula. Or, you expand everything by Taylor up to order eight. In any case, the computation fills several pages. There must surely be a geometrical reasoning to compute that limit.

[admissionsguy]

https://mathoverflow.net/questions/20696/a-question-regardin...

[GPerson]

You can at least guess the answer must be 1 instantly based on the idea that tan(x) and sin(x) equal x up to first order, and so therefore their inverses do too.

(So the limit is essentially (x - x)/(x - x) )

To make this rigorous do as you say about expanding the series.

Edit: Up to beyond first order actually*

[enriquto]

> the limit is essentially (x - x)/(x - x)

I agree with the intuition, but this is not so simple. A limit such as yours can still be anything, can't it? It depends on the higher-order terms. For example, the limit ((x+5x^2)-(x+2x^2))/((x+7x^2)-(x+6x^2)) is also "essentially" (x-x)/(x-x), but it turns out to be 3, not 1. In the given example, there are cancellations of even higher order terms, that you have to check that they cancel correctly. Following the links on the answer by admissionguy I found two nice arguments for the computation, an algebraic and a geometric one.