# The sandwich theorem.

Theorem 4.6.1   Let , and be three functions defined on the same neighborhood of some real number (resp. of , resp. of ). Suppose that:
(i)
For any , the double inequality holds.
(ii)
The functions and have the same limit at (resp. at , resp. at ).
Then has a limit at (resp. at , resp. at ) and it is equal to .

Example 4.6.2   Take , and . For any real number in , we have .

Moreover, and have both a positive infinite limit at , thus has a positive infinite limit at (see Figure fig example sandwich 1).

Corollary 4.6.3   Suppose that is bounded in a (pointed) neighborhood of and that . Then .

Example 4.6.4   We know that the sine function is bounded on , thus the function is bounded on any pointed neighborhood of 0. Moreover . Thus . See Figure fig example sandwich 2.

Noah Dana-Picard 2007-12-28