问题及解答

求

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

\[
\lim_{x\rightarrow\infty}\Bigl(\frac{x^2+x+1}{x^2+x}\Bigr)^{2x^2}=\lim_{x\rightarrow\infty}\Bigl[\bigl(1+\frac{1}{x^2+x}\bigr)^{x^2+x}\Bigr]^{\frac{2x^2}{x^2+x}}=e^2.
\]