$$L=\lim\limits_{x\to \infty}\left[\left(\left(\frac{x+1}{x-1}\right)^x-e^2\right)\ x^2 \right]$$
$\displaystyle{\lim_ {x\to \infty}}\left(\frac{x+1}{x-1}\right)^x$ is $1^{\infty}$ form and limit is $e^2$ so $L$ becomes$0\over0$ form.
let $t=1/x$
$$L=\displaystyle{\lim_ {t\to 0^{+}}}\frac{\left[\left(\frac{1+t}{1-t}\right)^{1/t}-e^2\right]}{t^2}$$
How I do proceed now?