Category:Definitions/Limits of Real Functions

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This category contains definitions related to Limits of Real Functions.
Related results can be found in Category:Limits of Real Functions.


Let $\openint a b$ be an open real interval.

Let $c \in \openint a b$.

Let $f: \openint a b \setminus \set c \to \R$ be a real function.

Let $L \in \R$.


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.