Definition:Limit of Real Function/Intuition
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Definition
Though the founders of Calculus viewed the limit:
- $\ds \lim_{x \mathop \to c} \map f x$
as the behavior of $f$ as it gets infinitely close to $x = c$, the real number system as defined in modern mathematics does not allow for the existence of infinitely small distances.
But:
- $\exists L: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$
can be interpreted this way:
- You want to get very close to the value $c$ on the $\map f x$ axis.
- This degree of closeness is the positive real number $\epsilon$.
- If the limit exists, I can guarantee you that I can give you a value on the $x$ axis that will satisfy your request.
- This value on the $x$ axis is the positive real number $\delta$.
Sources
- For a video presentation of the contents of this page, visit the Khan Academy.