Calculating Euler Number limit Please, so far I did
$$\lim_{x\to +\infty}\left(\frac{x^2-x+1}{x+2}\right)^{\frac{1}{x-1}},$$ but I can write
$$\frac{x^2-x+1}{x+2}=1+\frac{x^2-2x-1}{x+2}=1+\frac{1}{\frac{x+2}{x^2-2x-1}}.$$
But
$$\lim_{x\to +\infty}\frac{x+2}{x^2-2x-1}=0,$$ so I can not use 
$$e =\lim_{N\to \infty}(1+\frac{1}{N})^N$$

Notice that for sufficiently large $x$,
\begin{align}
1 < \left(\frac{x^2-x+1}{x+2}\right)^\frac{1}{x-1}
    & = \left(x-3+\frac{7}{x+2}\right)^\frac{1}{x-1} \\
    & < (x-1)^\frac{1}{x-1}
\end{align}
which has a well-known limit as $x \to +\infty$.
