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