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

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

  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

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

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]$ .
\begin{figure}\mbox{\epsfig{file=IntegerPart.eps,height=4cm}}\end{figure}

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:

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.
\begin{figure}\mbox{\epsfig{file=NoLimit.eps,height=4cm}}\end{figure}

Noah Dana-Picard 2007-12-28