Definition:Limit of Real Function
Definition
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.
That is:
- For every (strictly) positive real number $\epsilon$, there exists a (strictly) positive real number $\delta$ 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.
It can directly be seen that this definition is the same as that for a general metric space.
Terminology
When:
- $\ds \lim_{x \mathop \to c} \map f x = L$
it is usual to say that:
- as $x$ tends to $c$
or:
- as $x$ approaches $c$
then $\map f x$ approaches or tends to $L$.
Notation
$\map f x$ tends to the limit $L$ as $x$ tends to $c$, is denoted:
- $\map f x \to L$ as $x \to c$
or
- $\ds \lim_{x \mathop \to c} \map f x = L$
The latter is voiced:
- the limit of $\map f x$ as $x$ tends to $c$.
Limit from the Left
Let $\openint a b$ be an open real interval.
Let $f: \openint a b \to \R$ be a real function.
Let $L \in \R$.
Suppose that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: b - \delta < x < b \implies \size {\map f x - L} < \epsilon$
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 $f$, less than $b$ but within $\delta$ of $b$, has an image within $\epsilon$ of $L$.
Then $\map f x$ is said to tend to the limit $L$ as $x$ tends to $b$ from the left, and we write:
- $\map f x \to L$ as $x \to b^-$
or
- $\ds \lim_{x \mathop \to b^-} \map f x = L$
This is voiced:
- the limit of $\map f x$ as $x$ tends to $b$ from the left
and such an $L$ is called:
- a limit from the left.
Limit from the Right
Let $\Bbb I = \openint a b$ 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 $L$.
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
- $\ds \lim_{x \mathop \to a^+} \map f x = L$
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 at Infinity
Limit at (Positive) Infinity
$L$ is the limit of $f$ at infinity if and only if:
- $\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x > c : \size {\map f x - L} < \epsilon$
This is denoted as:
- $\ds \lim_{x \mathop \to \infty} \map f x = L$
Limit at Negative Infinity
$L$ is the limit of $f$ at minus infinity if and only if:
- $\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x < c: \size {\map f x - L} < \epsilon$
This is denoted as:
- $\ds \lim_{x \mathop \to - \infty} \map f x = L$
Intuition
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 $\delta$.
Examples
Example: Identity Function with $1$ at $0$
Let $f$ be the real function defined as:
- $\map f x = \begin {cases} x & : x \ne 0 \\ 1 & : x = 0 \end {cases}$
Then:
- $\ds \lim_{x \mathop \to 0} \map f x = 0$
Example: $\sqrt x$ at $1$
- $\ds \lim_{x \mathop \to 1} \sqrt x = 1$
Example: $e^{-1 / \size x}$ at $0$
- $\ds \lim_{x \mathop \to 0} e^{-1 / \size x} = 0$
Example: $\map \sin {\dfrac 1 x}$ at $0$
Let:
- $\map f x = \map \sin {\dfrac 1 x}$
Then:
- $\ds \lim_{x \mathop \to 0} \map f x$
does not exist.
Example: $x \map \sin {\dfrac 1 x}$ at $0$
Let:
- $\map f x = x \map \sin {\dfrac 1 x}$
Then:
- $\ds \lim_{x \mathop \to 0} \map f x = 0$
Also see
- Results about limits of real functions can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): limit: 1. (of a function)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): limit: 1. (of a function)