How to prove that $$\lim_{x\rightarrow 0 }\frac{\sin\big(\tan(x)\big)- \tan\big(\sin(x)\big)}{\arcsin\big(\arctan(x)\big)-\arctan\big(\arcsin(x)\big) } = 1$$ First terms in Taylor expansion of the numerator and denominator is $-\frac{x^7}{30}$. So the direct proof is not beautiful and hard.
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1You really should use parantheses to remove any potential ambiguity. I'm assuming, for example, that $\sin{\tan{(x)}} = \sin{(\tan{(x)})}$. Is that right? – layman Jan 19 '15 at 14:53
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i can't read it not really – Dr. Sonnhard Graubner Jan 19 '15 at 14:56
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@user46944 you are right. But I thing to much parantheses is not beautiful) – user2109257 Jan 19 '15 at 14:56
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2@user2109257 It's much more important to be clear on what you mean than to write beautiful math. The real skill is writing beautiful math while being crystal clear in your meaning. – layman Jan 19 '15 at 14:58
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@user46944 "being crystal clear in your meaning" isn't it the definition itself of "beautiful maths"? :) – Surb Jan 19 '15 at 15:08
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@MartinR thanks a lot. Then I close the question. – user2109257 Jan 19 '15 at 15:25