Difference between “boundary” and “limit” [closed]

Is there a difference between the semantics of the two words boundary and limit?

Is it possible that only one of the two has an inclusive meaning regarding the set we want the limit/boundary of? Compare it to mathematics, where we speak of “open” and “closed” sets and the difference between maximum (= the largest value in the set) and supremum (= the least upper bound, must not be contained in the set).

Solution 1:

Strictly speaking, a boundary is a visible mark which shows or sets a bound or limit.

The distinction can be clearly seen in the historical development of the word, which was formed from bound (“limit”) plus -ary (“connected with, pertaining to”). (OEtmD)

However you do not normally see this distinction made between boundary and bound or limit except in formal, technical contexts such as the law. In common, idiomatic usage, boundary often means bound or limit.

The idiomatic usage of boundary is nothing new. Consider the lexicographer’s complaint from the 1828 Webster’s Dictionary:

This word is thus used as synonymous with bound. But the real sense is, a visible mark designating a limit. Bound is the limit itself or furthest point of extension, and may be an imaginary line; but boundary is the thing which ascertains the limit; terminus, not finis. … But the two words are, in ordinary use, confounded.

Here, from the Macmillan Dictionary, are two sentences in which boundary is used in a sense which is not easily interchangeable with limit:

• The boundary between fact and fiction in her writing is often blurred.
• The Una River forms a natural boundary between the two countries.

And here, from Shakespeare courtesy of the 1913 Webster's Revised Unabridged Dictionary, are three sentences in which limit is used in a poetic sense not easily interchangeable with boundary, to mean not the bound but the entire space or time delimited:

• The archdeacon hath divided it / Into three limits very equally.
• The dateless limit of thy dear exile.
• The limit of your lives is out.

Solution 2:

According to the Merriam-Webster dictionary a boundary is something that indicates a limit, not a limit in and of itself. This agrees with my intuitive understanding of the two. Boundary lines are lines that mark the limits of the field, etc.