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.

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:

$\exists L: \forall \epsilon > 0: \exists \delta > 0: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

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.