Why FSA can't determine palindrome?

A finite state automaton (FSA) cannot determine palindromes because it lacks the memory capacity to store an arbitrary number of characters. Palindromes require the ability to compare characters from both ends of a string, which may be far apart, necessitating a form of memory to keep track of the first half while processing the second half. FSAs operate with a limited number of states and cannot "remember" an unbounded amount of information, making it impossible for them to recognize palindromic structures. Thus, palindromes need more powerful computational models, such as pushdown automata or Turing machines, which have the necessary memory capabilities.