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by GPerson
1897 days ago
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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* |
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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.