Epsilon-delta definition of the limit

Definition

The epsilon-delta definition of a limit formalizes the idea of a function approaching a value as the input approaches some point. It uses two values, $\epsilon$ and $\delta$, to define this behavior precisely.

5 Must Know Facts For Your Next Test

1. The definition states that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$ then $|f(x) - L| < \epsilon$.
2. $c$ is the point at which the limit is being evaluated, and $L$ is the value that $f(x)$ approaches.
3. $\epsilon$ represents how close $f(x)$ needs to be to the limit $L$.
4. $\delta$ represents how close $x$ needs to be to the point $c$.
5. The precise definition helps prove limits rigorously and is foundational for understanding continuity.

Review Questions

Related terms

Limit:

The value that a function approaches as its input approaches some point.

Continuity:

A function is continuous at a point if it is defined at that point, its limit exists at that point, and its limit equals its value at that point.

$\lim_{x \to c} f(x) = L$: This notation signifies that as x approaches c, f(x) approaches L.