calculus (and mathematical analysis in general) and its formal definition is used in turn to define notions like continuity, derivatives, and integrals. (In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.) The concept of limit was informally introduced for functions by Newton and Leibniz, at the end of the 17th century, for building infinitesimal calculus. For sequences, the concept was introduced by Cauchy, and made rigorous, at the end of the 19th century by Bolzano and Weierstrass, who gave the modern ε-δ definition, which follows. Definition. Let be a real-valued function defined on We say that tends to as approaches , or that the limit of as approaches is if, for any , there exists such that for all , implies that . We write this symbolically as or as Intuitively, this definition can be thought of in the following way: We say that as , when, given any positive number , no matter how small, we can always find a , such that we can guarantee that and are less than apart, as long as (in the domain of ) is a real number that is less than away from but distinct from . The purpose of the last stipulation, which corresponds to the condition in the definition, is to ensure that does not imply anything about the value of itself. Actually, does not even need to be in the domain of in order for to exist. In a slightly different but related context, the concept of a limit applies to the behavior of a sequence when becomes large. Definition. Let be a real-valued sequence. We say that converges to if, for any , there exists a natural number such that implies that . We write this symbolically as or as if fails to converge, we say that diverges. Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence and term by function and value and natural numbers and by real numbers and , respectively) yields the definition of the limit of as increases without bound, notated . Reversing the inequality to gives the corresponding definition of the limit of as decreases without bound, Sometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant.