I'm always infuriated by texture aliasing, and this is a very refreshing read of the basics. Notice that in this article the anisotropy is linearized, and near the horizon of the sampled plane this linearization breaks down. It is actually very difficult to render an infinite plane without aliasing nor blur near the horizon!
Picture zooming out from a repeating image. Eventually an entire picture takes a single pixel then multiple images per pixel. You can pick one pixel from a full image but that's going to get aliasing effects, or you can use multiple but that just means blur.
Increase the # of samples to any finite number, and you get the same problem at a higher zoom level.
The problem is that it is very difficult to filter a generic texture so that it looks perfect near the horizon. For a particular simple texture such as the checkerboard you can compute the exact rendering explicitly. But I know of no method to do it in general.