How Far Momentum Runs

Momentum uses

$v_t = \beta v_{t-1} + \nabla L(\theta_t), \qquad \theta_{t+1} = \theta_t - \alpha v_t$

with $\beta = 0.9$ and $\alpha = 0.01$, starting from $v_0 = 0$.

A model is descending a long, almost perfectly straight slope, so the gradient stays pinned at $\nabla L = 3$ for a very large number of consecutive steps.

As $t$ grows, the size of one update $|\theta_{t+1} - \theta_t|$ approaches a limiting value. What is that value?