The sandwich theorem.

Theorem 4.6.1   Let $f$ , $g$ and $h$ be three functions defined on the same neighborhood $\mathcal{V}$ of some real number $x_0$ (resp. of $-\infty$ , resp. of $+\infty$ ). Suppose that:

(i)

For any $x \in \mathcal{V}$ , the double inequality $f(x) \leq g(x) \leq h(x)$ holds.

(ii)

The functions $f$ and $h$ have the same limit $L$ at $x_0$ (resp. at $-\infty$ , resp. at $+\infty$ ).

Then $g$ has a limit at $x_0$ (resp. at $-\infty$ , resp. at $+\infty$ ) and it is equal to $L$ .

Figure 9: The sandwich theorem for functions.

Example 4.6.2   Take $f(x)=x^2$ , $g(x)=x^2+\sin x / x$ and $h(x)=x^2+1/x$ . For any real number in $(0,+\infty)$ , we have $f(x) \leq g(x) \leq h(x)$ .

Moreover, $f$ and $h$ have both a positive infinite limit at $+\infty$ , thus $g$ has a positive infinite limit at $+\infty$ (see Figure fig example sandwich 1).

Corollary 4.6.3   Suppose that $f$ is bounded in a (pointed) neighborhood of $x_0$ and that $\underset{x \rightarrow x_0}{\lim} g(x)=0$ . Then $\underset{x \rightarrow x_0}{\lim} f(x)g(x)=0$ .

Example 4.6.4   We know that the sine function is bounded on $\mathbb{R}$ , thus the function $x \mapsto \sin \frac 1x$ is bounded on any pointed neighborhood of 0. Moreover $\underset{x \rightarrow 0}{\lim} x=0$ . Thus $\underset{x \rightarrow 0}{\lim} x \cdot \sin \frac 1x =0$ . See Figure fig example sandwich 2.