Instructions: Each problem is worth $10$ point.

$(1)$ Evaluate $\underset{(x,y)(2,2)}{lim}\frac{x+y-4}{\sqrt[\phantom{2}]{x+y}-2}.$

$(2)$ Show that the limit $\underset{(x,y)(1,-1)}{lim}\frac{xy+1}{x^2-y^2}$ does not exist.

$(3)$ Determine the domain and range of the function $f(x,y,z)=\sqrt[\phantom{2}]{1-x^2-y^2-z^2}.$