Squeeze Theorem

How to use the squeeze theorem to solve the given limit: theorem, 1 example, and its solution.

Theorem

Theorem

For the case
when g(x) is between f(x) and h(x),

if the limit of the outer functions, f(x) and h(x),
goes to α (constant),

then the inner function g(x)
also goes to α.

This is the squeeze theorem.

Example

Example

Solution

To use the squeeze theorem,
start from
-1 ≤ sin x ≤ 1.

Sine: Graph

The given expression is (sin x)/x.

So divide each side by x:
-1/x ≤ (sin x)/x ≤ 1/x.

This example is the case when x → ∞.
So x is plus.
So dividing each side by x
does not change the order of the inequality sign.

Linear Inequality (One Variable)

The given expression is a limit
as x → ∞.

So, for each side,
write limx → ∞.

As x → ∞,
-1/x goes to
-1/∞ = -0 = 0.

If the denominator get bigger,
then the whole fraction gets smaller.
So -1/∞ = -0.

write the inequality signs and the inner term.

As x → ∞,
1/x goes to
1/∞ = 0.

So the limits of the outer functions are both 0.

Then, by the squeeze theorem,
the limit of the inner function, (sin)/x, is 0.

So 0 is the answer.