Limit of a function in a point

Considering a function $f (x)$ we say that it has limit $L$ in a point $p$ if $f (x)$ takes values as closed to $L$ as we want, taking sufficiently close points to $p$ but different from $p$ . This concept is denoted as:

$lim_{x \to p} f (x) = L$

Example

Let's take the function $f (x) = \frac{x^{2} - 1}{x}$ . If we look for the limit of the function at point $x = 5$ , we will see:

$lim_{x \to 5} f (x) = lim_{x \to 5} \frac{x^{2} - 1}{x} = \frac{5^{2} - 1}{5} = \frac{24}{5}$

In this case, the function $f (x)$ coincides with its limit at point $x = 5$ .

It can seem that the function always coincides with its limit at any point, but this is not the case. In the following example we see a case in which the function does not coincide with its limit:

Example

$f (x) = {\begin{matrix} 1 si x \neq 0 \\ 3 si x = 0 \end{matrix}$

In this case we see that $f (0) = 3$ , but:

$lim_{x \to 0} f (x) = lim_{x \to 0} 1 = 1$

This means that close to $x = 0$ the function always takes value $1$ and, consequently, its limit is $1$ . However, the function at $x$ equals zero has value $3$ .

This example is a clear example of discontinuous function. The discontinuous functions are detected easily since in the discontinuity points, the limits and the function do not coincide.

Therefore, to make the limit of a function $f (x)$ at point $p$ involves seeing all the values of the function $f (x)$ when we are located very close to $x = p$ , but not exactly on $p$ .