14. Given the functions $f(x)=x-\ln x-2, g(x)=x \ln x+x$.
(1) Prove that $f(x)$ has a zero in the interval $(3,4)$;
(2) If $k \in \mathbf{Z}$, and $g(x)>k(x-1)$ for all $x>1$, find the maximum value of $k$.
(1) $f^{\prime}(x)=1-\frac{1}{x}=\frac{x-1}{x}$, then $f(x)$ is monotonically increasing on $(1,+\infty)$,

and $f(3)=1-\ln 30$, so $f(x)$ has a zero point in $(3,4)$.
(2) Since $x>1$, then $k<\frac{x \ln x+x}{x-1}$, let $\varphi(x)=\frac{x \ln x+x}{x-1}$,
$$
\varphi^{\prime}(x)=\frac{(2+\ln x)(x-1)-(x \ln x+x)}{(x-1)^{2}}=\frac{x-\ln x-2}{(x-1)^{2}},
$$

From (1) we know there exists $x_{0} \in(3,4)$ such that $\varphi^{\prime}\left(x_{0}\right)=0$, and $\varphi(x)_{\min }=\varphi\left(x_{0}\right)$ $=\frac{x_{0} \ln x_{0}+x_{0}}{x_{0}-1}=\frac{x_{0}\left(x_{0}-2\right)+x_{0}}{x_{0}-1}=x_{0} \Rightarrow k<x_{0}$ always holds, so the maximum value of $k$ is 3.