Side limits

We know that 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 $p$ . This means that we are approximating $x = p$ , but how? From the right? From the left? We are going to specify the limit definition:

Limit from the left of $f (x)$ in $x = p$ :

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

Limit from the right of $f (x)$ in $x = p$ :

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

And if these two limits coincide $(L^{-} = L^{+})$ , then we say that:

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

Example

Let's take the function $f (x) = {\begin{matrix} 0 si x < 2 \\ 1 si x \geq 2 \end{matrix}$ and we will look for the side limits at $x = 2$ .

Limit from the left:

$L^{-} = lim_{x \to 2^{-}} f (x) = lim_{x \to 2^{-}} 0 = 0$

Limit from the right:

$L^{+} = lim_{x \to 2^{+}} f (x) = lim_{x \to 2^{+}} 1 = 1$

and nevertheless, the function in $x = 2$ is $1$ .

When a limit is computed, what can occur is that a function increases a great deal and we go so far as to say that the value of a limit is infinite.

Let's remember that we symbolize infinity with the symbol: $\infty$ .