Squeeze Theorem
The Squeeze Theorem (also called the Sandwich Theorem or the Squeeze Play Theorem) is a relatively simple theorem that deals with calculus, specifically limits.
Contents
[hide]Theorem
Suppose is between and for all in a neighborhood of the point . If and approach some common limit as approaches , then .
Proof
If is between and for all in the neighborhood of , then either or for all in this neighborhood. The two cases are the same up to renaming our functions, so assume without loss of generality that .
We must show that for all there is some for which implies .
Now since , there must exist such that
Now let . If then
So . Now by the definition of a limit we get as desired.
Applications and examples
The Squeeze Theorem can be used to evaluate limits that might not normally be defined. An example is the function with thelimit $\lim_{x\to\0} f(x)=x^2 e^{\sin\frac{1}{x}}$ (Error compiling LaTeX. Unknown error_msg). The limit is not normally defined, because the function oscillates infinitely many times around 0, but it can be evaluated with the Squeeze Theorem as following. Create two functions, and . It is easy to see that around 0, the function in question is squeezed between these two functions, and the limit as both of these approach 0 is 0, so $\lim_{x\to\0} x^2 e^{\sin\frac{1}{x}}$ (Error compiling LaTeX. Unknown error_msg) is 0.