Consider kinR and a sequence $(x_n).$ Show that, if $k>0,$ then $limin{f}_{n+}(k{x}_n)=klimin{f}_{n+}{x}_n,$ and $,limsu{p}_{n+}(k{x}_n)=klimsu{p}_{n+}{x}_n$ and if then $limin{f}_{n+}(k{x}_n)=klimsu{p}_{n+}{x}_n,$ and $,\underset{?}{lim}su{p}_{n+}(k{x}_n)=klimin{f}_{n+}{x}_n.$