A Limited Company is a common term used to refer to a corporation, whether public or private.
Fixed assets, which are usually Capital Assets (Property, Plant and Equipment) and are recorded at historical cost less any depreciation. Fair market vale (FMV) is only used when a corporation decides to discontinue a portion of its operations and the assets are classified held-for-sale.
Historical cost is what's reported on the face of the financial statements but if FMV would be more useful to the users of the statements it can always be disclosed in the notes.