asymptotics – Behaviour of harmonic oscillator when elastic constant$to +infty$.

I have a silly question about asymptotics and physical meaning.
Suppose you have this equation here

$$
left{
begin{array}{l}
x” + frac{k}{m} = -g \
x(0) = x_0 \
x'(0) = v_0
end{array}
right.
$$

Assuming I did all my calculations right the solution is something like

$$
x(t) = left(x_0 – frac{m}{k}gright) cosleft(sqrt{frac{k}{m}}t right) + sqrt{frac{m}{k}}v_0 sinleft(sqrt{frac{k}{m}} tright) – frac{m}{k}g
$$

Now if I let $kto +infty$ I get $x(t) sim x_0 cosleft(sqrt{frac{k}{m}}t right)$ which is kind of unespected in some sense and I’d like to know if the interpretation I am giving is correct:

  1. If $x_0 = 0$ then the system is not affected by the gravity at all, initial velocity won’t matter and there’s only one dominant term in the asymptotic expansion.
  2. If $x_0 neq 0$ the system will oscillate indefinitely, this bit was not really expected.

I was expecting to see the system completely still no matter what with an extremely stiff spring.

Is my interpretation correct?