Six, the tangent line to the curve $f(x)=x^{2}$ at the point $(1,1)$ intersects the x-axis at the point $(\alpha, 0)$. Find $\lim _{n \rightarrow \infty} f(\alpha)$.
Six, solve $\because f^{\prime}(x)=n x^{0-1}$,
$\therefore f(x)$ at the point $(1,1)$ has the tangent line equation
$$
y-1=n(x-1) \text {. }
$$

Let $\mathrm{y}=0$, we get $\alpha=1-\frac{1}{\mathrm{n}}$.
Thus, $f(\alpha)=\alpha^{*}=\left(1-\frac{1}{n}\right)^{n}$.
$\therefore \lim _{n \rightarrow \infty} f(\alpha)=\lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n}=\frac{1}{e}$.