# Definition:Limit of Real Function

## Contents

## 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 $L \in \R$.

Suppose that:

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

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

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.

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

- $\map f x \to L$ as $x \to c$

or

- $\displaystyle \lim_{x \mathop \to c} \map f x = L$

This is voiced:

**the limit of $\map f x$ as $x$ tends to $c$**.

It can directly be seen that this definition is the same as that for a general metric space.

### 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

- $\displaystyle \lim_{x \mathop \to b^-} \map f x = L$

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

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

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 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:

- $\displaystyle \lim_{x \mathop \to \infty} \map f x = L$

### 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:

- $\displaystyle \lim_{x \mathop \to - \infty} \map f x = L$

## Intuition

Though the founders of Calculus viewed the limit:

- $\displaystyle \lim_{x \mathop \to c} \map f x$

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 $\delta$.

## Sources

- 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions - 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): $\S 8.3$

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