问题及解答

求极限

\[\lim_{x\rightarrow\infty}(\frac{2x+1}{2x-3})^{3x-1}.\]

\[
\begin{split}
&\lim_{x\rightarrow\infty}\biggl(\frac{2x+1}{2x-3}\biggr)^{3x-1}\\
=&\lim_{x\rightarrow\infty}\biggl(1+\frac{4}{2x-3}\biggr)^{3x-1}\\
=&\lim_{x\rightarrow\infty}\Biggl[\biggl(1+\frac{4}{2x-3}\biggr)^{\frac{2x-3}{4}}\Biggr]^6\cdot\biggl(1+\frac{4}{2x-3}\biggr)^{\frac{7}{2}}\\
=&\ e^6\ .
\end{split}
\]