Definition:Limit of Real Function

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.


 * LimitOfFunction.png

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.

Limits involving infinity

 * $\displaystyle \forall \epsilon \in \R_{>0} : \exists c : \forall x > c : | f(x) - L | < \epsilon \implies \lim_{x \to \infty} f(x) = L$


 * $\displaystyle \forall \epsilon \in \R_{>0} : \exists c : \forall x < c : | f(x) - L | < \epsilon \implies \lim_{x \to - \infty} f(x) = L$

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 $\delta$.