Definition:Limit of a Function
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Limit of a Function on a Metric Space
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
Let $c$ be a limit point of $M_1$.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ except possibly at $c$.
Let $L \in M_2$.
Then $f \left({x}\right)$ is said to tend to (or approach) the limit $L$ as $x$ tends to (or approaches) $c$, and we write:
- $f \left({x}\right) \to L$ as $x \to c$
or
- $\displaystyle \lim_{x \to c} f \left({x}\right) = L$
if the following equivalent conditions hold.
This is voiced:
- the limit of $f \left({x}\right)$ as $x$ tends to $c$.
Epsilon-Delta Condition
- $\forall \epsilon > 0: \exists \delta > 0: 0 < d_1 \left({x, c}\right) < \delta \implies d_2 \left({f \left({x}\right), L}\right) < \epsilon$
where $\delta, \epsilon \in \R$.
That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every point in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some point $L$ in the codomain of $f$.
Epsilon-Neighborhood Condition
- $\forall N_\epsilon \left({L}\right): \exists N_\delta \left({c}\right) \setminus \left\{{c}\right\}: f \left({N_\delta \left({c}\right) \setminus \left\{{c}\right\}}\right) \subseteq N_\epsilon \left({L}\right)$.
where:
- $N_\delta \left({c}\right) \setminus \left\{{c}\right\}$ is the deleted $\delta $-neighborhood of $c$ in $M_1$
- $N_\epsilon \left({L}\right)$ is the $\epsilon$-neighborhood of $L$ in $M_2$.
That is, for every $\epsilon$-neighborhood of $L$ in $M_2$, there exists a deleted $\delta$-neighborhood of $c$ in $M_1$ whose image is a subset of that $\epsilon$-neighborhood.
Real and Complex Numbers
As:
- The real number line $\R$ under the usual metric forms a metric space;
- The complex plane $\C$ under the usual metric forms a metric space;
the definition holds for sequences in $\R$ and $\C$.
However, see the definition of the limit of a real function below:
Limit of a Real Function
The concept of the limit of a real function has been around for a lot longer than that on a general metric space.
The definition for the function on a metric space is a generalization of that for a real function, but the latter has an extra subtlety which is not encountered in the general metric space, namely: the "direction" from which the limit is approached.
Limit from the Left
Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$.
Suppose that:
- $\exists L: \forall \epsilon > 0: \exists \delta > 0: b - \delta < x < b \implies \left|{f \left({x}\right) - L}\right| < \epsilon$
where $L, \delta, \epsilon \in \R$.
That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number in the domain of $f$, less than $b$ but within $\delta$ of $b$, has an image within $\epsilon$ of some real number $L$.
Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $b$ from the left, and we write:
- $f \left({x}\right) \to L$ as $x \to b^-$
or
- $\displaystyle \lim_{x \to b^-} f \left({x}\right) = L$
This is voiced:
- the limit of $f \left({x}\right)$ as $x$ tends to $b$ from the left.
Sometimes the notation $\displaystyle f \left({b^-}\right) = \lim_{x \to b^-} f \left({x}\right)$ is seen.
Limit from the Right
Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$.
Suppose that $\exists L: \forall \epsilon > 0: \exists \delta > 0: a < x < a + \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$
where $L, \delta, \epsilon \in \R$.
That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number in the domain of $f$, greater than $a$ but within $\delta$ of $a$, has an image within $\epsilon$ of some real number $L$.
Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $a$ from the right, and we write:
- $f \left({x}\right) \to L$ as $x \to a^+$
or
- $\lim_{x \to a^+} f \left({x}\right) = L$
This is voiced
- the limit of $f \left({x}\right)$ as $x$ tends to $a$ from the right.
Sometimes the notation $\displaystyle f \left({a^+}\right) = \lim_{x \to a^+} f \left({x}\right)$ is seen.
Limit
Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$ except possibly at some $c \in \left({a .. b}\right)$.
Suppose that:
- $\exists L: \forall \epsilon > 0: \exists \delta > 0: 0 < \left|{x - c}\right| < \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$
where $L, \delta, \epsilon \in \R$.
That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some real number $L$.
$\epsilon$ is usually considered as having the connotation of being small in magnitude, but this is a misunderstanding of its intent: the point is that (in this context) $\epsilon$ can be made arbitrarily small.
Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $c$, and we write:
- $f \left({x}\right) \to L$ as $x \to c$
or
- $\displaystyle \lim_{x \to c} f \left({x}\right) = L$
This is voiced:
- the limit of $f \left({x}\right)$ as $x$ tends to $c$.
It can directly be seen that this definition is the same as that for a general metric space.
Complex Analysis
The definition for the limit of a complex function is exactly the same as that for the general metric space.
