7. Given $A=\left\{x \mid k x^{2}-2 x+6 k<0, k \neq 0\right\}$
(1) If $A \subseteq(2,3)$, find the range of $k$;
(2) If $(2,3) \subseteq A$, find the range of $k$;
(3) If $A \cap(2,3) \neq \varnothing$, find the range of $k$.
7. Solution (1) From the case of $k x^{2}-2 x+6 k0$,

then $f(x)=\frac{2 x}{x^{2}+6}=\frac{2}{x+\frac{6}{x}}$ has a maximum value of $f(\sqrt{6})=\frac{\sqrt{6}}{6}$ and a minimum value of $f(2)=f(3)=$ $\frac{2}{5}$ on $[2,3]$,

and $f(x)$ is monotonically increasing on $(0, \sqrt{6}]$ and monotonically decreasing on $[\sqrt{6},+\infty)$, so if $A \subseteq(2,3)$, there must be $k \geqslant$ $\frac{2}{5}$
(2) According to the analysis of the first question, if $(2,3) \subseteq A$, then $k \leqslant \frac{2}{5}$ and $k \neq 0$.
(3) From the analysis of the first question, if $A \cap(2,3) \neq \varnothing$, then $k<\frac{\sqrt{6}}{6}$ and $k \neq 0$.