Definition:Limit of a Real Function

Definition

Limit from the Left

Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$.

Suppose that:

$\exists L: \forall \epsilon > 0: \exists \delta > 0: b - \delta < x < b \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$.

That is, for every real positive $\epsilon$ there exists a real 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 some real number $L$.

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

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

This is voiced:

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

Sometimes the notation $\displaystyle f \left({b^-}\right) = \lim_{x \to b^-} f \left({x}\right)$ is seen.

Limit from the Right

Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$.

Suppose that $\exists L: \forall \epsilon > 0: \exists \delta > 0: a < x < a + \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$.

That is, for every real positive $\epsilon$ there exists a real 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 some real number $L$.

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

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

This is voiced

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

Sometimes the notation $\displaystyle f \left({a^+}\right) = \lim_{x \to a^+} f \left({x}\right)$ is seen.

Limit

Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$ except possibly at some $c \in \left({a .. b}\right)$.

Suppose that:

$\exists L: \forall \epsilon > 0: \exists \delta > 0: 0 < \left|{x - c}\right| < \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$.

That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some real number $L$.

$\epsilon$ 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.

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 $\delta$.

(Now ask me for a smaller $\epsilon$, I dare you.)

Sources

• James M. Hyslop: Infinite Series (1942): $\S 4$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definitions $1.3.1, \ 1.3.3$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.3$
• For a video presentation of the contents of this page, visit the Khan Academy.