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On the payment page, can I limit the total number of registrations accepted for any given class I'm teaching?
You can set a maximum number of submissions in a form, please take a look at this guide: http://www.jotform.com/help/137-How-to-Limit-Submissions-in-a-Form
Your second question didn't come completely, but I assume that you want to set a limit date to accept submissions. If this is the case, you can also see how to do this on the guide above.
Let us know if you need further assistance.