# Definition:Limit of Real Function

## Contents

## 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 $L$.

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

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

Let $L \in \R$.

Suppose that:

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

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

That is, for every real 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 $f \left({x}\right)$ 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

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

This is voiced

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

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.

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

- 1942: James M. Hyslop:
*Infinite Series*: $\S 4$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.3$: Limits of functions: Definition $1.3.1$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 8.3$ - For a video presentation of the contents of this page, visit the Khan Academy.