本题目来源于试卷: Maclaurin Series，类别为 IB数学
[问答题]
The function f is den nap-at, f6p(fine6mqvsa2+ h f)ed by $f(x)=e^{\arctan x}$ .
1. Find the first two derivatives of f(x) and hence find the Maclaurin series for f(x) up to and including the $x^{2} $ term.
2. Show that the coefficient of $x^{3}$ in the Maclaurin series for f(x) is $-\frac{1}{6}$ .
3. Using the Maclaurin series for $\sin x $ and $\ln (2 x+1) $, find the Maclaurin series for $\sin (\ln (2 x+1)) $ up to and including the $ x^{3}$ term.
4. Hence, or otherwise, find $\lim _{x \rightarrow 0} \frac{f(x)-1}{\sin (\ln (2 x+1))} $.
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