Squeezing Theorem

Let there be two functions $f_-(x)$ and $f_+(x)$ such that $f(x)$ is ``squeezed'' between the two,

$$f_-(x)\leq f(x)\leq f_+(x).$$

If

$$r=\lim_{x\to a} f_-(x) =\lim_{x\to a} f_+(x),$$

then $\lim_{x\to a} f(x)=r$. In the above diagram the functions $f_-(x)=-x^2$ and $f_+(x)=x^2$ ``squeeze'' $x^2\sin(c x)$ at 0, so $\lim_{x\to 0} x^2\sin(cx)=0$. The squeezing theorem is also called the Sandwich Theorem.