<s>[INST] Prove that the limit as $(x,y,z)$ approaches $0$ of $(x^3+y^3+z^3)/(xyz)$ does not exist I am asked to show that 
$$
\lim_{(x,y,z)\to(0,0,0)}\frac{x^3+y^3+z^3}{xyz}
$$
does not exist. I know that I need to use $x=at$, $y=bt$, and $z=bt$ but I don't understand what I should do after inserting them into the function.
 [/INST] In
$\frac{x^3+y^3+z^3}{xyz}
$
let
$y = ax$
and
$z = bx$.
These values are on
a line through the origin.
It becomes
$\frac{x^3+(ax)^3+(bx)^3}{x(ax)(bx)}
=\frac{x^3+a^3x^3+b^3x^3}{abx^3}
=\frac{1+a^3+b^3}{ab}
$.
Therefore,
along this line,
the ratio depends
only on $a$ and $b$.
Therefore
the limit does not exist
because,
by choosing different
$a, b$ values,
the ration can take on
different values.
For example,
if $a=b=1$,
the ratio is $3$;
if $a=b=2$,
the ratio is $17/4$;
if $b=a$,
the ratio is $(1+2a^3)/a^2$;
and so on.
</s>