Squeeze Theorem Formula

Squeeze theorem is if g(x) <= f(x) <= h(x) near x = a, and _x a g(x) = _x a h(x) = L, then _x a f(x) = L.

The Formula

If g(x)≀f(x)≀h(x)g(x) \leq f(x) \leq h(x) and lim⁑xβ†’ag(x)=lim⁑xβ†’ah(x)=L\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then lim⁑xβ†’af(x)=L\lim_{x \to a} f(x) = L.

When to use: If ff is squeezed between two functions that both approach the same value LL, then ff has no choiceβ€”it must also approach LL. Like being caught between two walls closing in to the same point.

Quick Example

Find lim⁑xβ†’0x2sin⁑(1x)\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right).
Since βˆ’1≀sin⁑(1x)≀1-1 \leq \sin\left(\frac{1}{x}\right) \leq 1, we have βˆ’x2≀x2sin⁑(1x)≀x2-x^2 \leq x^2\sin\left(\frac{1}{x}\right) \leq x^2.
Both βˆ’x2β†’0-x^2 \to 0 and x2β†’0x^2 \to 0, so lim⁑xβ†’0x2sin⁑(1x)=0\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0

Notation

g(x)≀f(x)≀h(x)g(x) \leq f(x) \leq h(x) β€” gg is the lower bound, hh is the upper bound, and ff is squeezed between them.

What This Formula Means

If g(x)≀f(x)≀h(x)g(x) \leq f(x) \leq h(x) near x=ax = a, and lim⁑xβ†’ag(x)=lim⁑xβ†’ah(x)=L\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then lim⁑xβ†’af(x)=L\lim_{x \to a} f(x) = L.

If ff is squeezed between two functions that both approach the same value LL, then ff has no choiceβ€”it must also approach LL. Like being caught between two walls closing in to the same point.

Formal View

If βˆƒΞ΄0>0\exists \delta_0 > 0 such that g(x)≀f(x)≀h(x)g(x) \leq f(x) \leq h(x) for all xx with 0<∣xβˆ’a∣<Ξ΄00 < |x - a| < \delta_0, and lim⁑xβ†’ag(x)=lim⁑xβ†’ah(x)=L\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then lim⁑xβ†’af(x)=L\lim_{x \to a} f(x) = L.

Worked Examples

Example 1

easy
Use the squeeze theorem to find lim⁑xβ†’0x2sin⁑ ⁣(1x)\displaystyle\lim_{x \to 0} x^2 \sin\!\left(\frac{1}{x}\right).

Answer

00

First step

1
Since βˆ’1≀sin⁑ ⁣(1x)≀1-1 \leq \sin\!\left(\frac{1}{x}\right) \leq 1 for all xβ‰ 0x \neq 0:

Full solution

  1. 2
    Multiply by x2β‰₯0x^2 \geq 0: βˆ’x2≀x2sin⁑ ⁣(1x)≀x2-x^2 \leq x^2\sin\!\left(\frac{1}{x}\right) \leq x^2.
  2. 3
    lim⁑xβ†’0(βˆ’x2)=0\lim_{x\to 0}(-x^2) = 0 and lim⁑xβ†’0(x2)=0\lim_{x\to 0}(x^2) = 0.
  3. 4
    By the squeeze theorem: lim⁑xβ†’0x2sin⁑ ⁣(1x)=0\lim_{x\to 0} x^2\sin\!\left(\frac{1}{x}\right) = 0.
The function sin⁑(1/x)\sin(1/x) oscillates wildly near 0, so its limit doesn't exist alone. Multiplying by x2x^2 traps it between βˆ’x2-x^2 and x2x^2, both going to 0, so the product must also go to 0.

Example 2

hard
Use the squeeze theorem to prove lim⁑xβ†’0sin⁑xx=1\displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1.

Example 3

medium
Use the squeeze theorem to evaluate lim⁑xβ†’0x2sin⁑(1/x2)x\lim_{x\to 0} \dfrac{x^2 \sin(1/x^2)}{x}.

Common Mistakes

  • Choosing bounds with unequal limits - the two bounding functions must approach the same LL or the theorem says nothing.
  • Bounding in the wrong direction - verify g(x)≀f(x)≀h(x)g(x)\le f(x)\le h(x) actually holds near aa, not just that the bounds look simpler.
  • Forgetting the inequality only needs to hold near aa - 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⁑xβ†’0sin⁑xx=1\lim_{x\to0}\frac{\sin x}{x}=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?" β€” rather than by familiar numbers β€” 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)g(x) \leq f(x) \leq h(x) near x=ax = a, and lim⁑xβ†’ag(x)=lim⁑xβ†’ah(x)=L\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then lim⁑xβ†’af(x)=L\lim_{x \to a} f(x) = L.

How do you use the Squeeze Theorem formula?

If ff is squeezed between two functions that both approach the same value LL, then ff has no choiceβ€”it must also approach LL. 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(x) \leq f(x) \leq h(x) β€” gg is the lower bound, hh is the upper bound, and ff 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⁑xβ†’0sin⁑xx=1\lim_{x\to0}\frac{\sin x}{x}=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?" β€” rather than by familiar numbers β€” 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 β†’
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