  In this paper, we study 3-dimensional complete non-compact Riemannian
manifolds with asymptotically nonnegative Ricci curvature and a uniformly
positive scalar curvature lower bound. Our main result is that, if this
manifold has $k$ ends and finite first Betti number, then it has at most linear
volume growth, and furthermore, if the negative part of Ricci curvature decays
sufficiently fast at infinity, then we have an optimal asymptotic volume ratio
$\limsup_{r\rightarrow\infty}\frac{\mathrm{Vol}(B(p, r))}{r}\leq4k\pi$. In
particular, our results apply to 3-dimensional complete non-compact Riemannian
manifolds with nonnegative Ricci curvature and a uniformly positive scalar
curvature lower bound.
