# Squeeze Theorem

How to solve limit problems by using the squeeze theorem: theorem, example, and its solution.

## Theorem

For *f*(*x*) ≤ *g*(*x*) ≤ *h*(*x*),

if the limit of *f*(*x*) and the limit of *h*(*x*) are both *L*

(as *x* → *a*),

then the limit of *g*(*x*) as *x* → *a* is *L*.

This is true because

as *x* → *a*,*L* ≤ [the limit of *g*(*x*)] ≤ *L*.

So [the limit of *g*(*x*)] is squeezed by those *L*-s.

This is why it is called the squeeze theorem

(aka the sandwich theorem).

## Example

Recall that -1 ≤ sin *x* ≤ 1.

Graphing sine functions

Divide both sides by *x*.

(It's doable because *x* → ∞ means *x* ≠ 0.)

The orders of the inequality signs don't change,

because *x* → ∞ means *x* is (+).

As *x* → ∞,

-1/*x* → 0 and 1/*x* → 0.

Then, by the squeeze theorem,

the limit of (sin *x*)/*x* is 0.

(sin *x*)/*x* is squeezed between the 0-s.