Evaluate $\lim\limits_{n\rightarrow\infty}\left(\frac{n^2 + 1}{n^2 - 2}\right)^{n^2}$
$$\lim\limits_{n\rightarrow\infty}\left(\frac{n^2 + 1}{n^2 - 2}\right)^{n^2}$$
Trying to solve this. At first thought it was $1$, but in wolfram it's $e^3$. Thanks
Solution 1:
$$\lim_{n\to +\infty}\left(\frac{n^2+1}{n^2-2}\right)^{n^2}=\lim_{n\to +\infty}\left(\left(1+\frac{1}{\frac{n^2-2}{3}}\right)^{\frac{n^2-2}{3}}\right)^{\frac{3n^2}{n^2-2}}$$
$$=e^{\lim_{n\to +\infty}\frac{3n^2}{n^2-2}}=e^{\lim_{n\to +\infty}\left(3+\frac{6}{n^2-2}\right)}=e^3$$