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 $k$.