Method guideCalculus IIntermediate8 min read

How and when to use the Squeeze Theorem

Trap a difficult function between two easier functions that are forced to meet at the same limit.

g(x)\le f(x)\le h(x)

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Reviewed July 11, 2026

Use it whenStart here

Use the Squeeze Theorem when the target oscillates or resists direct algebra, but its size can be bounded by expressions with the same limit.

The theorem

If g(x) is less than or equal to f(x), and f(x) is less than or equal to h(x) near a, and both outer functions approach L, then f(x) must also approach L.

The inequalities need to hold in a deleted neighborhood of the point, not necessarily at the point itself.

\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L\Longrightarrow\lim_{x\to a}f(x)=L

Look for a bounded factor

Sine and cosine always stay between −1 and 1. If an oscillating factor is multiplied by something approaching zero, absolute values often create the two useful bounds immediately.

The method is less about finding two magical functions than about controlling magnitude.

Why matching outer limits matter

If the bounds approach different numbers, the target still has room to move. The theorem becomes decisive only when the interval between the bounds collapses to a single value.

Worked exampleControl an oscillating function

Evaluate x² sin(1/x) as x approaches zero.

-1\le\sin(1/x)\le1

Bound the oscillating factor.

-x^2\le x^2\sin(1/x)\le x^2

Multiply by the nonnegative shrinking factor.

\lim_{x\to0}(-x^2)=\lim_{x\to0}x^2=0

The outside functions meet.

Result

\boxed{0}

Watch for

Common mistakes

Using bounds that do not hold on both sides of the point.
Choosing outer functions with different limits.
Multiplying an inequality by a quantity of unknown sign without checking direction.

Keep

Three takeaways

Control magnitude when exact simplification is impossible.
Absolute values make oscillation easier to bound.
Both outer limits must meet.