A limit describes the value a function approaches as its input gets arbitrarily close to some point ā even if the function is never actually evaluated exactly at that point. Limits are the foundational concept of calculus, providing the rigorous basis for both derivatives and integrals.
Imagine walking halfway to a wall, then halfway again, then halfway again ā forever. You never quite touch the wall, but you get closer and closer, approaching it arbitrarily closely. That's the intuition behind a limit: it's the value something approaches, even if it never quite gets there.
Limits let mathematicians make precise sense of ideas that seem paradoxical at first ā like "instantaneous speed" (how fast something is moving at a single instant, when speed usually means distance over time) or "the exact area under a curved line" (when area usually means length times width for straight-sided shapes). Both of these ideas ā the derivative and the integral ā are built directly from limits.
The ancient Greek philosopher Zeno posed a puzzle: to walk across a room, you must first walk halfway, then half of the remaining distance, then half of that, forever ā infinitely many steps. How can you ever arrive? Limits resolve this: an infinite sum of ever-smaller distances (Ā½ + Ā¼ + ā + ...) can still add up to a finite total (in this case, exactly 1) ā you do arrive, and calculus explains precisely why.
We write lim(xāa) f(x) = L to mean: as x gets arbitrarily close to a (without necessarily equaling a), f(x) gets arbitrarily close to L.
lim(xāa) f(x) = L means: for every Īµ > 0, there exists Ī“ > 0 such that whenever 0 < |xāa| < Ī“, it follows that |f(x)āL| < Īµ.
In words: no matter how small a margin of error Īµ you demand around L, we can find a small enough neighbourhood Ī“ around a (excluding a itself) where f(x) stays within that margin. This definition, due to Cauchy (1821) and made fully rigorous by Weierstrass (1861), replaced centuries of intuitive but imprecise talk of "infinitely small quantities."
The left-hand limit lim(xāaā») f(x) considers only x approaching a from below; the right-hand limit lim(xāaāŗ) f(x) only from above. The (two-sided) limit exists if and only if both one-sided limits exist and are equal.
lim(xāā) f(x) = L means f(x) approaches L as x grows without bound. For example, lim(xāā) 1/x = 0.
If lim f(x) = L and lim g(x) = M (as xāa), then: lim[f(x)+g(x)] = L+M, lim[f(x)Ā·g(x)] = LM, and lim[f(x)/g(x)] = L/M (provided M ā 0). These laws allow complex limits to be broken into simpler pieces.
A function f is continuous at a if lim(xāa) f(x) = f(a) ā the limit exists, f is defined at a, and they agree. This is the rigorous definition behind the intuitive idea of a function whose graph has "no breaks or jumps."
Newton and Leibniz (1660sā1680s) developed calculus using intuitive "infinitesimal" quantities ā numbers smaller than any positive real number but not zero. This worked in practice but lacked logical rigour, famously criticised by Bishop Berkeley (1734) as "ghosts of departed quantities." Cauchy (1821) began the process of rigorising calculus using limits rather than infinitesimals; Weierstrass (1861) completed this with the modern epsilon-delta definition, removing the need for infinitesimals entirely.
In 1966, Abraham Robinson developed non-standard analysis, using mathematical logic (model theory) to construct a rigorous number system ā the hyperreal numbers ā that does contain genuine infinitesimals. This provides a logically sound way to use Leibniz's original intuitive approach, and the two frameworks (epsilon-delta and non-standard analysis) are provably equivalent in what they can prove about ordinary calculus.
The limit concept extends to sequences: lim(nāā) aā = L means the sequence aā, aā, aā, ... gets arbitrarily close to L. A sequence converges if such a limit exists; otherwise it diverges. The Cauchy criterion characterises convergence intrinsically (without reference to the limit value L): a sequence converges if and only if its terms eventually get arbitrarily close to each other (a Cauchy sequence) ā in a complete space like ā, this is equivalent to converging to some limit.
The epsilon-delta definition generalises naturally to any metric space: lim(xāa) f(x) = L means for every Īµ > 0 there is Ī“ > 0 such that d(x,a) < Ī“ implies d(f(x),L) < Īµ, where d is the relevant distance function. In general topological spaces (without a metric), limits are defined using open neighbourhoods instead, generalising the concept even further beyond spaces with a notion of distance.