One-sided limit at one point.

Definition 4.2.1  

1. We say that the function $f$ has a right limit $l$ at $x_0$ and we denote $\underset{x \underset{>}{\rightarrow} x_0}{\lim} f(x) = l$ if

   $$\forall \varepsilon >0, \exists \delta >0, \; \vert \; x_0 < x < x_0+\delta \Longrightarrow \vert f(x)-l\vert<\varepsilon$$

   
   

2. We say that the function $f$ has a left limit $l$ at $x_0$ and we denote $\underset{x \underset{<}{\rightarrow} x_0}{\lim} f(x) = l$ if

   $$\forall \varepsilon >0, \exists \delta >0, \; \vert \; x_0 - \delta < x < x_0 \Longrightarrow \vert f(x)-l\vert<\varepsilon$$

   
   

We define the ``integer part function'':

Definition 4.2.2   For any real number $x$ , there exists a unique integer $n$ such that $n \leq x < n+1$ . The integer $n$ is called the integer part of $x$ and is denote by $[x]$ (v.i. Figure 6).

Let now $f(x)=[x]$ . Then: $\underset{x \underset{<}{\rightarrow} 1}{\lim} f(x) = 0$ and $\underset{x \underset{>}{\rightarrow} 1}{\lim} f(x) = 1$ .

Figure: The graph of $x \mapsto [x]$ .

Therefore the integer part function has no limit at 0, by Proposition 1.2. Similar definitions can be written for infinite one-sided limits. Please try to do it.

For example, if $f(x)=\frac 1x$ , then $\underset{x \underset{<}{\rightarrow} 0}{\lim} f(x) = -\infty$ and $\underset{x \underset{>}{\rightarrow} 0}{\lim} f(x) = +\infty$ .

Proposition 4.2.3   The function $f$ has a limit at $x_0 \in \mathbb{R}$ if, and only if, the three following conditions are fulfilled:

• $f$ has a left limit $l_1$ at $x_0$ ;
• $f$ has a right limit $l_2$ at $x_0$ ;
• $l_1=l_2$ .

For example, let $f(x)=\frac {x}{\vert x\vert}$ . Then $\underset{x \underset{<}{\rightarrow} 0}{\lim} f(x) = -1$ and $\underset{x \underset{>}{\rightarrow} 0}{\lim} f(x) = 1$ . Hence, $f$ has no limit at 0 (see Figure  7).

Figure 7: No limit at 0.