This is actually what the definition is which tests whether a function is continuous at c. In other words, I’d you want to know if f(x) is continuous at c, then the limit of x, as x approaches c, must be f(c).
Furthermore, this must remain true as x approaches c from both negative infinite and positive infinite. Said slightly differently, the limit as x approaches c from negative infinite can be f(c) while x approaches c from positive infinite is not f(c), and vice versa. It is important to check both directions.