how can I find the dpi of a pdf file in Preview?

I don't see a way to do this in Preview.

Acrobat Pro can tell you. I made a test PDF out of InDesign with two images. One at 300 dpi, and the other at 72 dpi. Opening the file and then choosing Preflight, you can list the page objects under PDF analysis.

Double click and it gives you a report. Below it shows me there are two images, their position, size and resolutions.

For much better control, there's the Acrobat Pro add-on, PitStop Pro. It's really expensive, but tells you much more. You can click on any object, and unlike Acrobat Pro's report above, it will even give you the file name.

I don't know if the free Acrobat Reader DC will do any of the above. I also looked around the web a bit and couldn't find any free method to get such information from the embedded objects of a PDF.

AFAIK PDFs don't really have a single DPI value as such. Raster/Bitmap objects have their own resolution and vector images such as fonts don't have any resolution at all. Inspecting a PDF in Preview I can't see anything relating to DPI so perhaps you can't? Adobe builds into some of its Apps a 'Preflight' tool and I think a DPI value should show up there when creating a PDF.

Perhaps others may know more and hopefully contribute.

Good luck

Read the information at How to determine the dpi of a PDF. Hint: Not without Adobe Acrobat Pro.