本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
The function f is defc3ch2 rexm e3+08 a3wy0ja ysdinedu uw(bfl i8:xfqhe/ /8bh8 *st 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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