Cramer's Rule is the method to solve for the discriminant of a non 2x2 matrix.
We could perhaps try it without the Hospital rule.
Let $\displaystyle t=\frac{1}{1-x}, \;\ x=1-\frac{1}{t}$
Then we get the recognizable and fairly famous limit which can be solved by a suitable substitution:
$\displaystyle \lim_{t\to {\infty}}\left(1-\frac{1}{t}\right)^{t}=\frac{1}{e}$
Probably wasn't worth posting, but I had to post at least one thing today