Squeeze theorem

In calculus, the squeeze theorem, (also known as the pinching theorem) is a theorem regarding with the limit of a function. It states that:

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

$\lim _{x\to a}f(x)=\lim _{x\to a}h(x)=L$

then $\lim _{x\to a}g(x)=L$.

Example

Consider g(x) = x² 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) = -x² and h(x) = x², 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)

$\lim _{x\to 0}f(x)=\lim _{x\to 0}g(x)=0\qquad \mathrm {(*)}$

We then have

$\lim _{x\to 0}g(x)=0$

because of (*).

Proof

It is given that

$\lim _{x\to a}f(x)=\lim _{x\to a}h(x)=L$

so by the definition of the limit of a function at a point, for any ε > 0 there is a δ₁ > 0 such that

if 0 < |x - a| < δ₁ then |f(x) - L| < ε

if 0 < |x - a| < δ₁ then -ε < f(x) - L < ε

if 0 < |x - a| < δ₁ then L - ε < f(x) < L + ε

and a δ₂ > 0 such that > 0 there is a δ₁ > 0 such that

if 0 < |x - a| < δ₂ then |h(x) - L| < ε and

if 0 < |x - a| < δ₂ then L - ε < h(x) < L + ε.

Then let δ equal the less of δ₁ and δ₂ (δ = min(δ₁, δ₂) ). 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 $\lim _{x\to a}g(x)=L$.