Limit
Calculus & Advanced MathA 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, what value does the function zoom in on? You never actually reach that input value; you just get infinitely close.
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 $6$.
Key Insight
Limits let us talk about what a function is "heading toward" even when it cannot actually reach that spot.
Definition
The limit of $f(x)$ as $x$ approaches $a$ equals $L$ if $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ (but not equal to $a$). Written $\lim_{x \to a} f(x) = L$.
Example
$\lim_{x \to 2} (x^2 - 4)/(x - 2)$: direct substitution gives $0/0$, but factoring gives $(x+2)(x-2)/(x-2) = x+2$, so the limit is $4$.
Key Insight
A limit can exist even if $f(a)$ is undefined. The function's value AT $a$ is irrelevant; only the behavior NEAR $a$ matters.
Definition
Formally (epsilon-delta): $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \epsilon$. This definition underpins all of analysis.
Example
Prove $\lim_{x \to 3} 2x = 6$: given $\epsilon > 0$, set $\delta = \epsilon/2$. Then $|x-3| < \delta$ implies $|2x-6| = 2|x-3| < 2(\epsilon/2) = \epsilon$.
Key Insight
The $\epsilon$-$\delta$ definition makes "approaching" rigorous, replacing intuition with a precise, quantified challenge-response game that forms the foundation of real analysis.