When "which" is the object of a preposition (to which, of which, for which, etc), the "which" must still replace the noun (or noun phrase) that precedes it.

Correct: She wants to borrow money, of which I don't have much.

In the given sentence, the phrase "the objectives of which" should modify the noun directly before, but it's not the money that has objectives - it's the limitations. We need a different kind of modifier here.

Here are all of the answer choices, with explanations:

Most states impose limitations on the authority of the legislature to borrow money, with their objectives being to protect taxpayers and the credit of the state government.

A. to borrow money, with their objectives being to protect taxpayers
B. to borrow money, the objectives of which are the protecting of
C. to borrow money, limitations intended to protect
D. for borrowing money, of which the objective is protecting
E. for borrowing money, limitations with the intent of protecting

A. to borrow money, with their objectives being to protect taxpayers
"with objectives being" is not idiomatically correct.

B. to borrow money, the objectives of which are the protecting of
As stated above, "of which" is nonsensically modifying "money." Also, "the protecting of" is not idiomatic. "The protection of" or "to protect" would be preferred.

C. to borrow money, limitations intended to protect
Correct. "Limitations" correctly refers to the limitations mentioned earlier, and "to protect" is idiomatically correct.

D. for borrowing money, of which the objective is protecting
The correct idiom is "authority TO" not "authority FOR." The "of which" was also incorrectly modifying "money."

E. for borrowing money, limitations with the intent of protecting
"for" is not idiomatic.

The correct answer is C.