I have a 21.5 inch iMac, late 2013. It runs very slowly. It came with 8GB of memory (two 4GB cards.) Can I upgrade those cards, or is that is all this machine will take? The cards are not user-accessible so I would have to take it to a professional.

+1 Run EtreCheck.

EtreCheck is a useful app to diagnose and identify issues, both hardware and software. It is trusted by many users. Please follow the steps outlined below:

1. Navigate to: https://etrecheck.com and get the free version of EtreCheckPro.
2. Download it. It is a free and trusted app. It is often recommended here on Apple Support Communities. 
3. Open EtreCheck, and before running the report, be sure to check “Allow full Drive Access” at the bottom of the EtreCheckPro window.
4. Run the report, and put it in a post, as seen here - https://discussions.apple.com/docs/DOC-250000211. We will then examine the report for issues in your Mac.

Cheers,

Jack

Find your model on https://everymac.com/ which will also tell you what upgrades are possible.

Note that a major contributor to a slow Mac of that era is its hard drive which was often a 5400 (vs. 7200) rotational drive. If somebody is doing brain surgery on your Mac have them replace it with a SSD. Alternatively, buy an external SSD and use that as a boot drive. Even over USB3 it may well be faster than your slow internal boot 5400 rpm HDD. You could even investigate that as a cheaper alternative to brain surgery. It won't get you the RAM boost but you may find unless you're doing major video production that you can live with 8GB. Your computer is 7 years old which puts it in retirements years for computers.

No, RAM in the 2012-current 21.5" iMacs is not user upgradeable. However, RAM is likely not the cause of a slow iMac, on a 2013 21.5" iMac it may be that you are still using a HD instead of a SSD. If you can do the following we will try to help you diagnose what you can do to speed up the computer.

Please navigate to www.Etrecheck.com and download the free version of EtreCheck. Once you have you downloaded the app and installed it, please run the report and save it. This report will help us get a good idea what has been installed on your system and help us be able to diagnose what may be wrong.

When you have your report, you can attach it when you reply to this message and we can then review it and help you determine what is needed to get your system running well again.

For instructions on how to download your EtreCheck report and attach it to your reply to this message please click https://discussions.apple.com/docs/DOC-250000211