问题及解答

求极限

\[
\lim_{x\rightarrow 0}\dfrac{\int_0^x\arctan(x-t)dt}{\sin(3x)\ln(1+2x)}
\]

\[
\begin{split}
\text{原式}&=\lim_{x\rightarrow 0}\frac{\int_0^x\arctan(x-t)dt}{3x\cdot 2x}\\
&\stackrel{u=x-t}{=}\lim_{x\rightarrow 0}\frac{\int_0^x\arctan udu}{6x^2}\\
&\stackrel{\text{洛}}{=}=\lim_{x\rightarrow 0}\frac{\arctan x}{12x}\\
&=\lim_{x\rightarrow 0}\frac{x}{12x}\\
&=\frac{1}{12}.
\end{split}
\]