Prove this limits with $\sin{(\tan{x})}-\tan{(\sin{x})}$

Question

How Find limit $$\lim_{x\to 0}\dfrac{\sin{(\tan{(\sin{(\tan{x})})})}-\tan{(\sin{(\tan{(\sin{x})})})}} {\sin{(\tan{x})}-\tan{(\sin{x})}}$$

My approach is the following: I use wolframalpha found this limits is $2$

Answer 1

For this kind of problems, Taylor series are very useful. Just start from the series of the most inner terms, cascade (replacing and simplifying) and be patient !$$\tan(x)=x+\frac{x^3}{3}+\frac{2 x^5}{15}+\frac{17 x^7}{315}+O\left(x^9\right)$$ $$\sin(\tan(x))=x+\frac{x^3}{6}-\frac{x^5}{40}-\frac{55 x^7}{1008}+O\left(x^9\right)$$ $$\tan(\sin(\tan(x)))=x+\frac{x^3}{2}+\frac{11 x^5}{40}+\frac{571 x^7}{5040}+O\left(x^9\right)$$ $$\sin(\tan(\sin(\tan(x))))=x+\frac{x^3}{3}+\frac{x^5}{30}-\frac{9 x^7}{70}+O\left(x^9\right)$$ Similarly $$\sin(x)=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+O\left(x^9\right)$$ $$\tan(\sin(x))=x+\frac{x^3}{6}-\frac{x^5}{40}-\frac{107 x^7}{5040}+O\left(x^9\right)$$ $$\sin(\tan(\sin(x)))=x-\frac{x^5}{10}-\frac{x^7}{63}+O\left(x^9\right)$$ $$\tan(\sin(\tan(\sin(x))))=x+\frac{x^3}{3}+\frac{x^5}{30}-\frac{13 x^7}{210}+O\left(x^9\right)$$ So the numerator is $$\sin(\tan(\sin(\tan(x))))-\tan(\sin(\tan(\sin(x))))=-\frac{x^7}{15}+O\left(x^9\right)$$ and the denominator is $$\sin(\tan(x))-\tan(\sin(x))=-\frac{x^7}{30}+O\left(x^9\right)$$ from which $$\lim_{x\to 0}\dfrac{\sin{(\tan{(\sin{(\tan{x})})})}-\tan{(\sin{(\tan{(\sin{x})})})}} {\sin{(\tan{x})}-\tan{(\sin{x})}}=2$$ Being much more patient and using an extra term for the expansions, you could show that $$\dfrac{\sin{(\tan{(\sin{(\tan{x})})})}-\tan{(\sin{(\tan{(\sin{x})})})}} {\sin{(\tan{x})}-\tan{(\sin{x})}}=2+\frac{5 x^2}{3}+O\left(x^3\right)$$ which shows the limit and how it is approached.

Edit

I forgot to precise that it is a good parctice to start investigating the simplest term; here, it is the denominator. Its expansion tells basically the degree of what you should do with the numerator.

Since the mechanism is in place, you could show for the next level $$\frac{\sin (\tan (\sin (\tan (\sin (\tan (x))))))-\tan (\sin (\tan (\sin (\tan (\sin (x))))))}{\sin (\tan (\sin (\tan (x))))-\tan (\sin (\tan (\sin (x))))}=\frac{3}{2}+\frac{5 x^2}{4}+O\left(x^3\right)$$

Added later

If we define $S(x)=\sin(\tan(x))$ and $T(x)=\tan(\sin(x))$, the expression given in the post is $$A_1=\frac{S(S(x))-T(T(x))}{S(x)-T(x)}$$ and the last given is $$A_2=\frac{S(S(S(x)))-T(T(T(x)))}{S(S(x))-T(T(x))}$$ What is interesting is that the asymptotic development of $A_n$ is simply given by $$A_n=\frac{n+1}{n}\big(1+\frac 56 x^2\big)$$ Amazing, isn't it ?