Limits

Definition

A limit is the value that a function “approaches” as the input approaches some value. It is the fundamental concept upon which all of calculus is built.

Why It Matters

Limits are the “atoms” of calculus. Without them, we couldn’t define derivatives (instant change) or integrals (accumulation); they are the logical foundation that allows us to mathematically handle the infinite and the infinitesimal.

Core Concepts

• Formal Definition: $\lim_{x \to c} f(x) = L$ means that as $x$ gets arbitrarily close to $c$, $f(x)$ gets arbitrarily close to $L$.

  • How to read: “The limit of f of x as x approaches c equals L.”
  • Meaning / when to use: Describes intended output near $c$ without requiring $f(c)=L$ or even that $f(c)$ exists.

• Approaching vs. Being: A limit cares about the behavior near a point, not the value at the point. A function can have a limit at $c$ even if $f(c)$ is undefined.

• Foundation of calculus: Derivatives and Integrals are both defined using limits.

Connected Concepts

• Calculus
• Continuity at a Point
• Differential calculus
• Convergence
• Divergence