Definition:Limit of Real Function

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Definition

Limit from the Left

Let $\left({a \,.\,.\, b}\right)$ be an open real interval.

Let $f: \left({a \,.\,.\, b}\right) \to \R$ be a real function.

Let $L \in \R$.


Suppose that:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: b - \delta < x < b \implies \left\vert{f \left({x}\right) - L}\right\vert < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

That is, for every real strictly positive $\epsilon$ there exists a real strictly 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 $L$.


LimitFromLeft.png

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

$\displaystyle \lim_{x \mathop \to b^-} f \left({x}\right) = L$


This is voiced:

the limit of $f \left({x}\right)$ as $x$ tends to $b$ from the left.

and such an $L$ is called:

a limit from the left.


Limit from the Right

Let $\Bbb I = \openint r s$ be an open real interval.

Let $f: \Bbb I \to \R$ be a real function.

Let $L \in \R$.


Suppose that:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \Bbb I: a < x < a + \delta \implies \size {\map f x - L} < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

That is, for every real strictly positive $\epsilon$ there exists a real strictly 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 $L$.


LimitFromRight.png

Then $\map f x$ is said to tend to the limit $L$ as $x$ tends to $a$ from the right, and we write:

$\map f x \to L$ as $x \to a^+$

or

$\displaystyle \lim_{x \mathop \to a^+} \map f x = L$


This is voiced

the limit of $\map f x$ as $x$ tends to $a$ from the right

and such an $L$ is called:

a limit from the right.


Limit

Let $\left({a \,.\,.\, b}\right)$ be an open real interval.

Let $c \in \left({a \,.\,.\, b}\right)$.

Let $f: \left({a \,.\,.\, b}\right) \setminus \left\{{c}\right\} \to \R$ be a real function.

Let $L \in \R$.


Suppose that:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \left\vert{x - c}\right\vert < \delta \implies \left\vert{f \left({x}\right) - L}\right\vert < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number $x \ne c$ in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of $L$.


$\epsilon$ 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.


LimitOfFunction.png

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.


Limit at Infinity

$L$ is the limit of $f$ at infinity if and only if:

$\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x > c : \left\vert{f \left({x}\right) - L}\right\vert < \epsilon$

This is denoted as:

$\displaystyle \lim_{x \mathop \to \infty} f \left({x}\right) = L$


$L$ is the limit of $f$ at minus infinity if and only if:

$\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x < c : \left\vert{f \left({x}\right) - L}\right\vert < \epsilon$

This is denoted as:

$\displaystyle \lim_{x \mathop \to - \infty} f \left({x}\right) = L$


Intuition

Though the founders of Calculus viewed the limit:

$\displaystyle \lim_{x \to c} f \left({x}\right)$

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:

$\exists L: \forall \epsilon > 0: \exists \delta > 0: 0 < \left|{x - c}\right| < \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

can be interpreted this way:

You want to get very close to the value $c$ on the $f\left({x}\right)$ 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