Squeeze Theorem

Summary

The squeeze theorem (sandwich theorem) finds a limit by trapping a function between two others that share the same limit.

Prerequisites

Limits

Definition / Theorem

Suppose $g(x)\le f(x)\le h(x)$ for all $x$ in a deleted neighborhood of $a$ . If

then

The same idea works for one-sided limits and for $x\to \pm \mathrm{\infty }$ .

Conditions / Assumptions

• The inequality must hold near the limit point (except possibly at the point itself).
• Both bounding functions must share the same finite (or infinite) limit. One bound alone is not enough.

Worked Example

Because $-1\le \sin (1/x)\le 1$ for $x\ne 0$ ,

Since $\underset{x\to 0}{lim}(-|x|)=\underset{x\to 0}{lim}|x|=0$ , the squeeze theorem gives

Common Mistakes

• Using only an upper bound (for example $0<e^{-x^2}<x^4$ ) and concluding a wrong limit. In fact $\underset{x\to \mathrm{\infty }}{lim}{e}^{-x^2}=0$ .
• Applying the theorem when the two bounds have different limits.

Connections

• Related: Fundamental Limits of Calculus
• Uses: many trigonometric limits

References

The squeeze theorem is a standard limit theorem in calculus.^[1]