Limits
- The ε–δ challenge-and-response definitionnot yet tested
- From infinitesimals to Weierstrass rigornot yet tested
- Approaching infinity and long-run behaviornot yet tested
- Continuity, derivatives, and integrals as limitsnot yet tested
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.