This is an ODE problem,but I cannot visualize why it can go to infinity or negative infinity.
$$x'' -6x' + 8x = 0$$
Where $x''$ is acceleration, $-6x'$ is the damping effect and $8x$ is the spring effect.
If I write it back into $physics$ form, I get $x'' = 6x' - 8x$. This just means the spring wants to pull the mass back and the resistive force is not strong enough.
Now the original question was, "Using the mass–spring analogy, predict the behavior as $t \to \infty$ of the solution to the given initial value problem. Then confirm your prediction by actually solving the problem."
Solving using Maple 15, I got (with Initial conditions)
I don't understand how infinity could be at play for this. How could you even have negative infinity? The spring wouldn't stretch that long and wouldn't the "max" point of the position be where the spring is attached (if it does go to negative infinity)