Squeeze theorem
In
calculus, the
squeeze theorem, (also known as the
pinching theorem) is a
theorem regarding the
limit of a function. This theorem
argues that if two functions approach the same limit at a point, and a third function "lies" betwixt those functions; then, the third function also approaches that limit at that point.
If the functions f, g, and h are defined in an interval I containing a except possibly at a itself, and f(x) ≤ g(x) ≤ h(x) for every number x in I for which x ≠ a, and
then .
Consider g(x) = x^{2} sin 1/x.
Trying to calculate the limit of g as x → 0 is difficult by conventional means; substitution will fail since we have a 1/x in the function. Trying to use L'Hôpital's rule fails too; it does not remove the 1/x term. So we turn to using this result.
Let f(x) = x^{2} and h(x) = x^{2}, these constitute lower and upper bounds to g(x) and satisfy then f(x) ≤ g(x) ≤ h(x).
We trivially have (because f and h are polynomials)

We then have
because of (*) and the squeeze theorem.
Proof
It is given that
so by the definition of the limit of a function at a point, for any
ε > 0 there is a
δ_{1} > 0 such that
 if 0 < x  a < δ_{1} then f(x)  L < ε
 if 0 < x  a < δ_{1} then ε < f(x)  L < ε
 if 0 < x  a < δ_{1} then L  ε < f(x) < L + ε
and a
δ_{2} > 0 such that
> 0 there is a δ
_{1}'' > 0 such that
 if 0 < x  a < δ_{2} then h(x)  L < ε and
 if 0 < x  a < δ_{2} then L  ε < h(x) < L + ε.
Then let
δ equal the less of
δ_{1} and
δ_{2} (
δ = min(
δ_{1},
δ_{2}) ). From the previous statements it follows that
 if 0 < x  a < δ then L  ε < f(x) and
 if 0 < x  a < δ then h(x) < L + ε.
It is given that
f(
x) ≤
g(
x) ≤
h(
x), so
 if 0 < x  a < δ then L  ε < f(x) ≤ g(x) ≤ h(x) < L + ε.
 if 0 < x  a < δ then L  ε < g(x) < L + ε.
 if 0 < x  a < δ then ε < g(x)  L < ε.
 if 0 < x  a < δ then g(x)  L < ε.
This fits the definition of a limit for the function
g as
x approaches
a, so
.