Equivalence of Definitions of Limit of Real Function
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Theorem
The following definitions of the concept of Limit of Real Function are equivalent:
Definition 1
$\map f x$ tends to the limit $L$ as $x$ tends to $c$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$
where $\R_{>0}$ denotes the set of strictly positive real numbers.
Definition 2
$\map f x$ tends to the limit $L$ as $x$ tends to $c$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: x \in \map {N_\delta} c \setminus \set c \implies \map f x \in \map {N_\epsilon} L$
where:
- $\map {N_\epsilon} L$ denotes the $\epsilon$-neighborhood of $L$
- $\map {N_\delta} c \setminus \set c$ denotes the deleted $\delta$-neighborhood of $c$
- $\R_{>0}$ denotes the set of strictly positive real numbers.
Proof
By definition of deleted $\delta$-neighborhood of $c$:
- $x \in \map {N_\delta} c \setminus \set c$
- $0 < \size {x - c} < \delta$
By definition of $\epsilon$-neighborhood of $L$:
- $\map f x \in \map {N_\epsilon} L$
- $\size {\map f x - L} < \epsilon$
The result follows by comparison of the definitions.
$\blacksquare$