问题及解答

求极限

\[\lim\limits_{n\rightarrow\infty}\Bigl(1+\frac{1}{n}+\frac{1}{n^2}\Bigr)^n\].

\[
\begin{split}
&\lim_{n\rightarrow\infty}\Bigl(1+\frac{1}{n}+\frac{1}{n^2}\Bigr)^n\\
=&\lim_{n\rightarrow\infty}\Bigl(1+\frac{n+1}{n^2}\Bigr)^n\\
=&\lim_{n\rightarrow\infty}\Bigl(1+\frac{n+1}{n^2}\Bigr)^{\frac{n^2}{n+1}\cdot\frac{n+1}{n}}\\
=&\lim_{n\rightarrow\infty}e^{\frac{n+1}{n}\ln(1+\frac{n+1}{n^2})^{\frac{n^2}{n+1}}}\\
=&e^1\\
=&e.
\end{split}
\]