Here is a rough sketch, which can be made rigorous by somebody determined: the
$n$-nacci numbers, as the solutions of a linear difference equation, can be expressed as linear combinations of powers of the roots of the corresponding characteristic polynomial.
To give a more concrete example, let's look at the usual Fibonacci sequence (and is easily generalized to the higher-order versions). Since we have the relation
$$f_{n}-f_{n-1}-f_{n-2}=0$$
then the characteristic polynomial is
$$x^2-x-1$$
(notice the correspondence?), whose roots are
$\phi$ and
$-\frac1{\phi}$. So, the Fibonacci (and thus the Lucas numbers as well) are expressible as
$f_n=A\phi^n+B\left(-\frac1{\phi}\right)^n$
where constants
$A$ and
$B$ can be determined from the initial conditions given. In particular, these characteristic polynomials have only one positive root
$\chi$ (
$\chi=\phi$ in the Fibonacci case), so asymptotically
$f_n\approx\chi^n$. So, the limit you are taking is effectively
$$\lim_{n\to\infty}\frac{f_{n+1}}{f_n}=\lim_{n\to\infty}\frac{\chi^{n+1}}{\chi^n}=\chi$$
