Squeeze Theorem Formula

Squeeze Theorem: if g(x) ≤ f(x) ≤ h(x) near x = a, and lim_(x → a) g(x) = lim_(x → a) h(x) = L, then lim_(x → a) f(x) = L.

The Formula

If g(x)≤f(x)≤h(x) and lim⁡x→ag(x)=lim⁡x→ah(x)=L, then lim⁡x→af(x)=L.

When to use: If f is squeezed between two functions that both approach the same value L, then f has no choice—it must also approach L. Like being caught between two walls closing in to the same point.

Quick Example

Find lim⁡x→0x2sin⁡(1x).
Since −1≤sin⁡(1x)≤1, we have −x2≤x2sin⁡(1x)≤x2.
Both −x2→0 and x2→0, so lim⁡x→0x2sin⁡(1x)=0

Notation

g(x)≤f(x)≤h(x) — g is the lower bound, h is the upper bound, and f is squeezed between them.

What This Formula Means

If g(x)≤f(x)≤h(x) near x=a, and lim⁡x→ag(x)=lim⁡x→ah(x)=L, then lim⁡x→af(x)=L.

If f is squeezed between two functions that both approach the same value L, then f has no choice—it must also approach L. Like being caught between two walls closing in to the same point.

Formal View

If ∃δ0>0 such that g(x)≤f(x)≤h(x) for all x with 0<∣x−a∣<δ0, and lim⁡x→ag(x)=lim⁡x→ah(x)=L, then lim⁡x→af(x)=L.

Worked Examples

Example 1

easy
Use the squeeze theorem to find lim⁡x→0x2sin⁡ ⁣(1x).

Answer

0

First step

1
Since −1≤sin⁡ ⁣(1x)≤1 for all x≠0:

Full solution

  1. 2
    Multiply by x2≥0: −x2≤x2sin⁡ ⁣(1x)≤x2.
  2. 3
    lim⁡x→0(−x2)=0 and lim⁡x→0(x2)=0.
  3. 4
    By the squeeze theorem: lim⁡x→0x2sin⁡ ⁣(1x)=0.
The function sin⁡(1/x) oscillates wildly near 0, so its limit doesn't exist alone. Multiplying by x2 traps it between −x2 and x2, both going to 0, so the product must also go to 0.

Example 2

hard
Use the squeeze theorem to prove lim⁡x→0sin⁡xx=1.

Example 3

medium
Use the squeeze theorem to evaluate lim⁡x→0x2sin⁡(1/x2)x.

Common Mistakes

  • Choosing bounds with unequal limits - the two bounding functions must approach the same L or the theorem says nothing.
  • Bounding in the wrong direction - verify g(x)≤f(x)≤h(x) actually holds near a, not just that the bounds look simpler.
  • Forgetting the inequality only needs to hold near a - it does not have to hold everywhere, just in a neighborhood of the point.

Why This Formula Matters

It is the standard escape hatch when direct substitution and algebra fail on oscillating or bounded-times-shrinking expressions, and it is how the foundational limit lim⁡x→0sin⁡xx=1 is proved. It trains the powerful habit of solving a hard limit by comparison rather than computation. Recognizing it by "Can I bound this function between two functions that approach the SAME limit at the point?" — rather than by familiar numbers — is what lets a student tell it apart from direct substitution and l'hopital's rule and intermediate value theorem in a mixed problem set.

Frequently Asked Questions

What is the Squeeze Theorem formula?

If g(x)≤f(x)≤h(x) near x=a, and lim⁡x→ag(x)=lim⁡x→ah(x)=L, then lim⁡x→af(x)=L.

How do you use the Squeeze Theorem formula?

If f is squeezed between two functions that both approach the same value L, then f has no choice—it must also approach L. Like being caught between two walls closing in to the same point.

What do the symbols mean in the Squeeze Theorem formula?

g(x)≤f(x)≤h(x) — g is the lower bound, h is the upper bound, and f is squeezed between them.

Why is the Squeeze Theorem formula important in Math?

It is the standard escape hatch when direct substitution and algebra fail on oscillating or bounded-times-shrinking expressions, and it is how the foundational limit lim⁡x→0sin⁡xx=1 is proved. It trains the powerful habit of solving a hard limit by comparison rather than computation. Recognizing it by "Can I bound this function between two functions that approach the SAME limit at the point?" — rather than by familiar numbers — is what lets a student tell it apart from direct substitution and l'hopital's rule and intermediate value theorem in a mixed problem set.

What do students get wrong about Squeeze Theorem?

The procedure for squeeze theorem is the easy part; the trap is choosing bounds with unequal limits. Asking "Can I bound this function between two functions that approach the SAME limit at the point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Squeeze Theorem formula?

Before studying the Squeeze Theorem formula, you should understand: limit.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Limits Explained Intuitively: The Foundation of Calculus →
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