$($a$)$ Consider the function with formula

$f(x,y)=\sqrt[\phantom{2}]{x^2{y}^2-4}$

$($i$)$ Find and describe the domain of $f.$

$[5$ marks$]$

$($ii$)$ Find and describe the level curves of the graph of $f,$ at the level $c,$ for any value of cinR.

$[10$ marks$]$

$($b$)($i$)$ Show that the limit

$\underset{(x,y)(0,0)}{lim}\frac{x\sqrt[\phantom{2}]{y}}{(x-2\sqrt[\phantom{2}]{y}{)}^2}$

does not exist.

$[10$ marks$]$

$($ii$)$ Show that

$\underset{(x,y,z)(0,0,0)}{lim}\frac{xyz}{x^2{y}^2{z}^2}=0.$

$[10$ marks$]$