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
- The definition states that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$ then $|f(x) - L| < \epsilon$.
- $c$ is the point at which the limit is being evaluated, and $L$ is the value that $f(x)$ approaches.
- $\epsilon$ represents how close $f(x)$ needs to be to the limit $L$.
- $\delta$ represents how close $x$ needs to be to the point $c$.
- The precise definition helps prove limits rigorously and is foundational for understanding continuity.
Review Questions
- What do the symbols $\epsilon$ and $\delta$ represent in the epsilon-delta definition?
- How does one use the epsilon-delta definition to prove that a function has a particular limit?
- What role does the value of $c$ play in this definition?
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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.
