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Squeeze theorem

In calculus, the squeeze theorem or sandwich theorem permits the evaluation of a limit by "squeezing" the function between two other functions which share a limit.[cite:@openstax2020calculus][cite:@wikipedia2025squeeze]

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sin x over x

graphically derive $\sin θ \leq θ \leq \tan θ$ then $\frac{\sin θ}{\sin θ} \leq \frac{θ}{\sin θ} \leq \frac{t a n θ}{\sin θ}$ THEN $1 \leq \frac{θ}{\sin θ} \leq \frac{1}{\cos θ}$ THENNN $c o s θ \leq \frac{\sin θ}{θ} \leq 1$ ok now $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ woohoo!

now do $\lim_{u \to 0} \frac{\sin u}{\frac{u}{6}}$ and you will freak out

Statement

Let $f x$ , $g x$ , and $h x$ be defined for all $x \neq a$ over an open interval containing $a$ . If $f x ⩽ g x ⩽ h x$ for all $x \neq a$ over the interval, and $\lim_{x \to a} f x = \lim_{x \to a} h x = L,$ then $\lim_{x \to a} g x = L$ .

The functions $f$ and $h$ are said to be lower and upper bounds of $g$ .