[tex]\sqrt{5+x}[/tex] is defined as long as [tex]5+x\ge0[/tex], or [tex]x\ge-5[/tex]. However, [tex]\dfrac1x[/tex] is undefined when [tex]x=0[/tex]. This point is contained in the set [tex]\{x|x\ge-5\}[/tex], so we need to exclude it. We can write the domain in several ways.
[tex]\{x|x\ge-5~\land~x\neq0\}[/tex]
[tex]-5\le x<0~\land~0<x<\infty[/tex]
[tex][-5,0)\cup(0,\infty)[/tex]
