A second

The limit of printer resolution is how small the steps of the paper and the nozzle carriage can be. Printing at 5000 dpi on the long axis would require the steps of paper movement to be 5 microns. They aren't and to the best of my knowledge, they can't be. The drops coming out of the nozzles can be as small as you like, which is why you can have so many dpi, but that isn't increasing resolution.

The limit of human vision is about 10 lp/mm, given excellent lighting and high contrast - and no refractive error - so 5-6 lp/mm for a photograph is plenty.

I never said that the printer 5000 dpi along the long axis; in the 1440x5760 figure, the first figure is along the direction of roller movement, the second along the direction of lateral movement of the print head (the relationship to long and short dimensions are unclear if roll paper is used). Since the printer lays out ink over a width of the paper in one pass using a matrix of nozzles and then moves the paper a bit and lays out another layer of ink with partial overlap, I'm not sure if it is necessary to move the paper in steps corresponding to the resolution of the image details along the long dimension of the paper, it could lay out finer details than the roller step size by using the nozzle matrix - at least in theory. I don't know if this is the case though.

Here is some testing of the print resolution at different input resolutions (and printer settings), using Epson Ultra Luster Paper:

https://www.printerknowledge.com/threads/star-chart-resolution-tests-of-the-epson-p900.16920/

The authors conclude that "From these tests, I concluded that, for my needs, using 720 Image PPI and the 1440x1440 print quality setting was a good choice. This setting provided a decent increase in resolution while enabling the full P900 resolution at the cost of 4X the image pixels. I would use 720 Image PPI and the 1440x5760 print quality setting if, in rare cases, the image justified a slight resolution or edge quality boost. The small increment in resolution for a portion of the angular space with an Image PPI of 1440 did not seem worth another 4X total image pixels."

720 ppi = 28 pixels per mm which can reproduce at most 14 lpmm was concluded a good choice for the source image, although in some cases a higher resolution of the source image (1440 ppi) produced slightly more resolution along some angular directions, but this was not considered worthwhile by the author. But clearly if we are talking about the limitations of the printer, the source material should be of the highest quality.

Glossy paper is likely to be somewhat better in reproducing fine detail than luster, but I haven't used it on the P900 and that was not tested in the above article.

Even if some print-making techniques produce more detailed or sharper small prints than a P900, for me this has not been an issue. Having the fast turnaround (a few min) and control afforded by having all the equipment to edit the image and make prints in one place along with free choice of print size (up to A2 in my case), the high maximum density of inkjet prints, and excellent light permanence of the prints means I basically never source out printing of my images, though I recommend people to have commercial labs do the thank-you cards because type C paper has better handling and wet resistance than the inkjet prints so they are more likely to survive handling by the postal service. If I need to make larger than A2 prints, then of course I source out, but that's not typically the case. I've never been happy with the print to screen matching of commercial labs and I've had so many quality issues with them that I can't imagine making prints if I had to outsource them on a regular basis. It would throw away any joy I have from photography and cause so much frustration.

Shop, shop

I have noticed over ≈15 years of inkjet printing at home with Epson pigment printers that, at least under my command, the inkjet printers cannot reproduce the sense of "sharpness" and detail one sees in a 8x12cm image printed in a good photo book made by conventional lithography.  There seems to be limit to inkjet that is exceeded by other systems.  Another example of a medium that exceeds inkjet would be contact prints from 8x10 inch film. The smaller the inkjet print, the more obvious the detail deficit is, assuming the original image has lots of detail.

Nature's expressions can be quite poetic. Snow writing belongs there.

Quintessential Birna!