Option 11: Squeeze Theorem (Sandwich Theorem)

Definition: Squeeze Theorem (aka Sandwich Theorem)

If  $\mathit{g}\mathit{\left(}\mathit{x}\mathit{\right)}\mathit{\le }\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}\mathit{\le }\mathit{h}\mathit{\left(}\mathit{x}\mathit{\right)}$ for all $\mathit{x}$ in some open interval containing $\mathit{c}$.

and

$\underset{x\to c}{\lim }\mathit{ }\mathit{g}\mathit{\left(}\mathit{x}\mathit{\right)}\mathit{=}\underset{x\to c}{\lim }\mathit{ }\mathit{h}\mathit{\left(}\mathit{x}\mathit{\right)}\mathit{=}\mathit{L}$.

Then you know that    $\underset{x\to c}{\lim }\mathit{ }\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}\mathit{=}\mathit{L}$

Note:

–          Essentially what the Squeeze Theorem is saying is that if you have an equation, $\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}$, that is ALWAYS between two other equations, $\mathit{g}\mathit{\left(}\mathit{x}\mathit{\right)}$ and $\mathit{h}\mathit{\left(}\mathit{x}\mathit{\right)}$, in some x-interval , AND those two equations, $\mathit{g}\mathit{\left(}\mathit{x}\mathit{\right)}$ and $\mathit{h}\mathit{\left(}\mathit{x}\mathit{\right)}$, have the same limit value , L , at the same x-value = c , THEN $\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}$ also has the same limit value, L , at that same x-value = c .

–          In other words, $\mathit{g}\mathit{\left(}\mathit{x}\mathit{\right)}$ and $\mathit{h}\mathit{\left(}\mathit{x}\mathit{\right)}$ squeeze together to make $\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}$ have the same limit value , L . You could also say that $\mathit{g}\mathit{\left(}\mathit{x}\mathit{\right)}$ and $\mathit{h}\mathit{\left(}\mathit{x}\mathit{\right)}$ sandwich $\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}$ between them, forcing it to have the same limit value , L .

–          The Squeeze Theorem is not a commonly applied method due to the challenge of finding equations that are always above and below a third equation. There are some very specific uses of it in an AP Calculus course, but this should not be a limit option you immediately look to apply.