If an offer was made on a house in which the buyer intended to have a hair salon in the basement, but found out a few days prior to closing that the zoning wouldn't allow it, can they cancel the agreement without penalty?

Yes, but only if they terminate in writing prior to the Record Title Objection Deadline

Explanation:

Zoning is a matter of Title. Buyers have the right to terminate the contact without penalty if any aspect of Title is unsatisfactory to them. However, they must due so in writing prior to either the Off-Record Title Objection Deadline or the Record Title Objection Deadline.

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