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 $L \in \R$.

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.


 * LimitOfFunction.png

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.

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 $\delta$.