Newton and Leibniz invented calculus in the 1660s and 1680s using a notion that did not, strictly, exist. Infinitesimals — quantities smaller than any positive number but not zero — gave the right answers and embarrassed the philosophers. Bishop George Berkeley mocked them in 1734 as "the ghosts of departed quantities," and his criticism was correct: nobody could say precisely what an infinitesimal was. The repair took a hundred and fifty years. Cauchy (1820s), Bolzano, and finally Karl Weierstrass in the 1850s replaced the ghost with a discipline: the epsilon-delta definition of a limit. Calculus became, for the first time, a subject in which every step could be checked.
The idea: a function ƒ has limit L as x approaches a if, for every positive ε no matter how small, there exists a positive δ such that whenever 0 < |x − a| < δ, we have |ƒ(x) − L| < ε. Translated: however close to L you want the output to be, you can guarantee it by making the input close enough to a. The infinitesimal is gone; what remains is a challenge-and-response — give me your tolerance ε, I will give you a margin δ that works. From this single mechanism the rest of analysis unfolds. Continuity at a point is exactly the statement that the limit equals the value: lim x→a ƒ(x) = ƒ(a). The derivative is the limit of a difference quotient: lim h→0 [ƒ(x+h) − ƒ(x)] / h. The integral is the limit of Riemann sums as the partition gets finer. Infinite series converge to a sum if and only if their partial sums have a limit. Asymptotic behavior — limits at infinity — captures long-run growth. The unifying observation is that every continuous process in mathematics is, formally, a limit. The ε–δ discipline is what lets a proof actually be a proof rather than a story about ghosts. The cost of the rigor is real: a first calculus course in the 1700s could be done with intuition, while a first analysis course today is largely a course in becoming fluent in this challenge-and-response. The benefit is that everything since 1860 is checkable in a way that everything before was not.
Limits are everywhere computation needs to know what happens eventually. Numerical analysis asks when an iterative algorithm converges and how fast. Asymptotic complexity in computer science (the O(n²) notation) is a limit statement about how a runtime scales as input grows. Statistical estimators are evaluated by their large-sample limits. Physics recovers classical mechanics from quantum mechanics by taking the limit ℏ → 0. Economics studies long-run equilibria — the limits of dynamic processes. Even category theory's notion of "limit" — at first glance unrelated — turns out to generalize the analytic one in a way that has reorganized large parts of modern mathematics.