Answer
(a)Substituting x=at, y=bt, z=ct in
$\lim\limits_{(x,y,z)\to (0,0,0)}$$\frac{xyz}{x^2+y^4+z^4}$ gives
$\lim\limits_{t\to 0}$$\frac{abct^3}{a^2t^2+b^4t^4+c^4t^4}$
taking $t^2$ common from above gives
$\lim\limits_{t \to 0}$$\frac{abct(t^2)}{t^2(a^2+b^4t^2+c^4t^2)}$ = $\frac{abc(0)}{a^2+b^4(0)^2+c^4(0)^2}$ = $\frac{0}{a^2}$ = 0
Work Step by Step
(b)Substituting x=$t^2$, y=t, z=t in
$\lim\limits_{(x,y,z)\to (0,0,0)}$$\frac{xyz}{x^2+y^4+z^4}$ gives
$\lim\limits_{t\to 0}$$\frac{(t^2)(t)(t)}{t^4+t^4+t^4}$ = $\lim\limits_{t \to 0}$$\frac{t^4}{3t^4}$ = $\frac{1}{3}$
From (a) and (b),
As $\lim\limits_{t \to 0}$, both the values aren't same, i.e. 0$\ne$$\frac{1}{3}$
Thus, limit Doesn't Exist as (x,y,z) approaches (0,0,0).
