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:

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

Let's take the function $f(x)=\left\{\begin{array}{c} 0 \ \text{ si } x < 2 \\ 1 \ \text{ si } x\geq2 \end{array} \right.$ 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: $\infty$.