How to use the squeeze theorem to solve the given limit: theorem, 1 example, and its solution.
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.
To use the squeeze theorem,
-1 ≤ sin x ≤ 1.
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.