Consider a particle in a three-dimensional potential $$V(x,y,z)=\frac{A \left(x^3+2 y^3+3 z^3+4 a^3\right)}{\left(x^2+y^2+z^2+a^2\right)^2}$$ This particle has scattering states if $E>E_0$, where
$E_0=$
How does one go about calculating the limit as $x,y,z$ approach infinity for a multivariable function? Take my photo for example.
Treat the variables as an $\mathbb{R}^n$ vector, and then extend that vector towards infinity: $$\lim_{x,y,z \to \infty} f(x,y,z) = \lim_{r \to \infty} f(C_x r, C_y r, C_z r), \quad 0 \ne C_x, C_y, C_z \in \mathbb{R}$$
In other words, $$\begin{aligned} V(x, y, z) & = \frac{ A ( x^3 + 2 y^3 + 3 z^3 + 4 a^3 ) }{( x^2 + y^2 + z^2 + a^2 )^2 } \\ \lim_{x,y,z \to \infty} V(x,y,z) & = \lim_{r \to \infty} V(X r, Y r, Z r) \\ ~ & = \lim_{r \to \infty} \frac{ A r^3 ( X^3 + 2 Y^3 + 3 Z^3 ) + 4 A a^3 ) }{( (X^2 + Y^2 + Z^2) r^2 + a^2 )^2 } = 0 \\ \end{aligned}$$
When you have a multivariate function, and wish to find the limit when two or more variables approach infinity, you should consider those $N$ variables as an $\mathbb{R}^N$ point, and all other variables constants.
Treat the direction of that point (with respect to origin) as a constant, and the distance from origin the only variable.
In the original question, we have three variables $x$, $y$, and $z$, that all approach infinity. We use three real constants, say $X$, $Y$, and $Z$ to describe the direction, and $r$ the single variable, so that $x \to r X$, $y \to r Y$, and $z \to r Z$. This way, we convert the multivariate limit to a limit in $r$ only.
If the direction constants ($X$, $Y$, and $Z$) do not appear in the resulting limit, we know the limit is not dependent on the direction. If they do, the limit is dependent on the direction, and the limit itself tells us how.
In that sense, the direction constants do not matter, because we expect them to not affect the limit; but we do need to use them in order to be sure. We never care about their numerical value (other than that they're never all zero, because that is not a valid direction; and we can pretty safely assume none of them are zero if that makes evaluating the limit easier).