Squeeze Theorem

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 goes to a), then the limit of g(x) as x goes to a is L.

For f(x) ≤ g(x) ≤ h(x),

if the limit of f(x) and the limit of h(x) are both L
(as xa),

then the limit of g(x) as xa is L.

This is true because
as xa,
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

Find the limit of the given expression. The limit of (sin x)/x as x goes to infinity.

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.