How to find the limit of sin x x as x approaches ∞ ?

Question

How to find the limit of                     sin        x            x       as x approaches       ∞  ?

Mia Small · Accepted Answer

We use squeeze theorem, which says that for three functions g(x), f(x), and h(x),
If

g (x) \leq f (x) \leq h (x), and lim_{x \to a} g (x) = lim_{x \to a} h (x) = L

then

lim_{x \to a} f (x) = L

.

lim_{x \to \pm \infty} \frac{\sin x}{x} = 0

Explanation:
Given that

f (x) = \frac{\sin x}{x}

We know that sin x oscillates between -1 and +1 for all values of x. Thus, if we set

g (x) = \frac{- 1}{x} and h (x) = \frac{+ 1}{x}

we have located the two functions satisfying the first condition that

g (x) \leq f (x) \leq h (x)

which can be written as
Given that

\frac{- 1}{x} \leq \frac{\sin x}{x} \leq \frac{+ 1}{x} for all values of x in (- \infty, \infty)

We know that

lim_{x \to \pm \infty} \frac{- 1}{x} = 0 and also that lim_{x \to \pm \infty} \frac{+ 1}{x} = 0

It follows from the squeeze theorem that

lim_{x \to \pm \infty} \frac{\sin x}{x} = 0

How do you find the limit of (sin x)/x as x approaches infty?