# 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?