Limit
A limit describes the value a function approaches as its input gets closer and closer to a specific number.
Formula
\lim_{x \to a} f(x) = L
Definition
A limit asks: as you slide $x$ closer and closer to some number $a$, what value does the function zoom in on, without $x$ ever actually reaching $a$? Formally, $\lim_{x \to a} f(x) = L$ means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ (but not equal to $a$). The rigorous epsilon-delta version makes this precise: for every $\epsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \epsilon$, a definition that underpins all of analysis.
Example
Imagine walking toward a wall: a limit asks how close you get, not whether you touch it. For $f(x) = 2x$, as $x$ gets close to $3$, $f(x)$ gets close to $6$. A trickier case is $\lim_{x \to 2} (x^2 - 4)/(x - 2)$: direct substitution gives $0/0$, but factoring gives $(x+2)(x-2)/(x-2) = x+2$, so the limit is $4$ even though the function itself is undefined at $x=2$. To prove $\lim_{x \to 3} 2x = 6$ rigorously: given $\epsilon > 0$, set $\delta = \epsilon/2$; then $|x-3| < \delta$ implies $|2x-6| = 2|x-3| < 2(\epsilon/2) = \epsilon$.
Key Insight
Limits let us talk about what a function is "heading toward" even when it cannot actually reach that spot, and a limit can exist even if $f(a)$ is undefined, since the function's value AT $a$ is irrelevant, only the behavior NEAR $a$ matters. The $\epsilon$-$\delta$ definition replaces this intuition with a precise, quantified challenge-response game that forms the foundation of real analysis.