Let $(M^3,g(t))$, $t \in [0,T)$, be a complete Ricci flow with bounded curvature on each compact time interval. Suppose that at $t=0$ the least curvature-operator eigenvalue satisfies $\nu(x,0) \geq -1$ for all $x \in M$. Then there are constants $s_0>0$ and a dimensional pinching function $\Phi:[s_0,\infty)\to [0,\infty)$ with $\Phi(s) \to 0$ as $s \to \infty$ such that, at every point with $R(x,t) \geq s_0$,

\begin{align*} \nu(x,t) \geq -R(x,t)\,\Phi(R(x,t)). \end{align*}

In particular, every blow-up limit at points with $R(x_k,t_k) \to \infty$ has nonnegative curvature operator.