# 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:

### 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:

## 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: $\dfrac 1 x$ at $+\infty$

- $\ds \lim_{x \mathop \to +\infty} \dfrac 1 x = 0$

### Example: $\dfrac {\sin x} x$ at $+\infty$

- $\ds \lim_{x \mathop \to +\infty} \dfrac {\sin x} x = 0$

alternating between positive and negative values.

### 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)