The Squeeze Theorem

The Central Idea

Suppose three functions satisfy

for \(x\) sufficiently close to \(a\), and suppose

and

Then the function in the middle must also approach \(L\):

Why the Theorem Works

The lower function \(g(x)\) approaches \(L\) from one side or nearby, while the upper function \(h(x)\) also approaches \(L\).

Since

the middle function has no room to approach a different value.

As the two outer functions close in on \(L\), the middle function is squeezed toward the same target.

The Formal Statement

Let \(f\), \(g\), and \(h\) be functions defined near \(a\), possibly not at \(a\) itself.

If

for all \(x\) sufficiently close to \(a\), and

then

The Middle Function Does Not Need an Easy Formula

The Squeeze Theorem is useful precisely because \(f(x)\) may be difficult to analyze directly.

We only need enough information to place it between functions whose limits are easier to evaluate.

A Simple Numerical Picture

Suppose near \(x=a\),

where

Then both bounds approach \(4\).

Therefore,

The Absolute-Value Version

One of the most useful forms of squeezing begins with an absolute-value bound.

If

and

then

Therefore,

Why Absolute Values Are So Useful

An absolute-value inequality controls both positive and negative behavior at once.

From

we immediately obtain

If the right-hand bound shrinks to zero, the middle expression is forced to zero.

A Basic Oscillatory Bound

The sine function satisfies

for every real \(t\).

Equivalently,

This simple fact becomes powerful when sine is multiplied by something that approaches zero.

Example: \(x\sin(1/x)\)

Evaluate

The factor

oscillates and has no limit as \(x\to0\).

But it remains bounded:

Finish the Squeeze

Multiply the bound by \(|x|\):

Since

we have

Therefore,

Oscillation Can Be Controlled by a Shrinking Factor

The expression

keeps oscillating.

But multiplying it by \(x\) makes the amplitude shrink toward zero.

The oscillation survives, but its vertical range collapses.

Another Example: \(x^2\cos(1/x)\)

Since

we have

Because

the Squeeze Theorem gives

A General Bounded-Factor Pattern

Suppose

near \(x=a\), where \(M\) is a fixed positive constant, and suppose

Then

Since

we obtain

Bounded Times Vanishing Goes to Zero

This produces a useful shortcut:

The bounded factor does not need to have a limit of its own.

Example with a Bounded Unknown Function

Suppose

near \(x=0\).

Evaluate

We have

Since

therefore

The Famous Trigonometric Limit

One of the most important uses of the Squeeze Theorem is proving

This limit later becomes fundamental in differentiating trigonometric functions.

A Geometric Comparison

For small positive \(x\), measured in radians, unit-circle geometry gives

Dividing by the positive quantity \(x\),

Rearrange the Trigonometric Inequality

Since

the inequality

implies

Multiplying by \(\cos x\) for sufficiently small positive \(x\),

Dividing by \(x>0\),

The Squeeze for \(\sin x/x\)

We now have

for sufficiently small positive \(x\).

As \(x\to0\),

and

Therefore,

Extend the Result to Both Sides

Because

the function

is even wherever it is defined.

Therefore the same limit holds from the left:

Radians Are Essential

The standard limit

assumes that \(x\) is measured in radians.

Using degrees changes the scale factor and therefore changes the limit.

A Related Limit

Evaluate

Rewrite:

Since

therefore

Another Related Limit

Evaluate

Rewrite:

Therefore,

A Derived Cosine Limit

Another important result is

One way to see this is to rationalize:

Rewrite as

The first factor approaches \(1\), and the second approaches \(0\).

Therefore the product approaches \(0\).

A Second Important Cosine Limit

We also have

Using

we obtain

The squared factor approaches \(1\), leaving

Squeezing Does Not Require Symmetric Bounds

The lower and upper functions do not need to look like

Any two functions work as long as

and both outer functions approach the same limit.

Example with Different Bounds

Suppose near \(x=2\),

As \(x\to2\),

and

Therefore,

One-Sided Squeeze Theorem

The theorem also works for one-sided limits.

If

for \(x\) approaching \(a\) from the right, and

then

Squeeze Theorem at Infinity

The same idea works as

or

If a function is trapped between two bounds that approach the same long-run value, the middle function has the same limit.

Example at Infinity

Evaluate

Since

we have

for \(x>0\).

Since

therefore

Another Limit at Infinity

Evaluate

Because

we get

Since

we conclude

The Bounding Functions Need Only Work Nearby

For a limit at \(x=a\), the inequality

does not need to hold for every real \(x\).

It only needs to hold sufficiently close to the target.

Limits are local statements.

The Function Need Not Be Defined at the Target

The Squeeze Theorem concerns nearby values.

Therefore, \(f(a)\) may be undefined and the theorem can still determine

This is consistent with the general theory of limits.

How to Find Useful Bounds

The hardest part is often discovering the inequalities.

Useful sources include:

• known trigonometric bounds such as \(|\sin x|\le1\);
• absolute-value inequalities;
• maximum and minimum values of bounded functions;
• simple comparisons of positive quantities;
• geometric inequalities;
• dominant-term estimates.

Start by Bounding the Difficult Part

If an expression contains an oscillatory or otherwise difficult factor, isolate that factor first.

For example, in

the difficult part is the sine factor.

But

Therefore,

Finish the Bound

Since

as \(x\to0\),

the Squeeze Theorem gives

A General Power-Oscillation Pattern

For any positive power \(p\),

as \(x\to0\).

Therefore, if \(g(x)\) is bounded near \(0\),

This pattern appears repeatedly in analysis.

Comparison with a Vanishing Envelope

It is often helpful to think of the outer bounds as an envelope around the difficult function.

If the envelope collapses toward one value, the function inside is forced to collapse toward the same value.

A Reliable Procedure

1. Identify the difficult part of the function.
2. Find a lower and upper bound, or an absolute-value bound.
3. Verify that the inequalities hold near the target.
4. Evaluate the limits of the bounding functions.
5. Check that both bounding limits agree.
6. Apply the Squeeze Theorem.
7. State the middle function's limit explicitly.

Example: Full Squeeze Workflow

Evaluate

Since

we have

Therefore,

Both outer functions approach \(0\).

Hence,

When the Squeeze Theorem Does Not Help

The method fails if the two bounding functions approach different values.

For example, if

the bounds approach \(0\) and \(1\), not the same number.

These inequalities alone do not determine the limit of \(f(x)\).

Bounding Alone Is Not Enough

The key requirement is not merely that a function is bounded.

The bounds must converge to the same limit.

A bounded oscillating function such as

remains between \(-1\) and \(1\), but those constant bounds do not squeeze toward one value.

Common Error: Reversing an Inequality Incorrectly

Multiplying or dividing an inequality by a negative quantity reverses the inequality signs.

This is one reason absolute-value bounds are often safer and cleaner.

Common Error: Using Bounds with Different Limits

The Squeeze Theorem requires

If the lower and upper limits differ, the theorem cannot identify a unique limit for the middle function.

Common Error: Forgetting to Verify the Inequality

A correct-looking pair of outer limits is not enough.

You must also know that

actually holds near the target.

Common Error: Assuming the Middle Function Needs Its Own Limit First

The whole purpose of the Squeeze Theorem is to determine the middle limit from the outer limits.

We do not assume the middle limit in advance.

Common Error: Treating Oscillation as Automatic Failure

Oscillation alone does not always destroy a limit.

The expression

has no limit as \(x\to0\).

But

does have the limit \(0\), because the oscillation is squeezed into a shrinking envelope.

Practice 1: Basic Absolute-Value Squeeze

Suppose

near \(x=0\).

Find

Practice 2: Oscillatory Product

Evaluate:

Practice 3: Higher Power

Evaluate:

Practice 4: Bounded Unknown Function

Suppose

near \(x=0\).

Evaluate:

Practice 5: Standard Trigonometric Limit

Evaluate:

Practice 6: Scaled Sine Limit

Evaluate:

Practice 7: Another Scaled Limit

Evaluate:

Practice 8: Limit at Infinity

Evaluate:

Practice 9: Compare the Bounds

Suppose

near \(x=1\).

Find

Practice 10: Identify the Error

A student argues:

and concludes

Explain why the Squeeze Theorem does not justify that conclusion.

Practice Answers

Practice 1

From

we get

Both bounds approach \(0\), so

Practice 2

Since \(|x|\to0\),

Practice 3

Since \(|x|^3\to0\),

Practice 4

Since \(12x^2\to0\),

Practice 5

Practice 6

Rewrite:

Therefore,

Practice 7

Therefore,

Practice 8

Since

we have

for \(x>0\).

Since \(1/x\to0\),

Practice 9

Both bounds approach \(2\):

and

Therefore,

Practice 10

The lower bound approaches \(-1\), while the upper bound approaches \(1\).

The bounding limits are not equal, so the Squeeze Theorem does not apply.

In fact,

oscillates and has no limit as \(x\to0\).

What to Remember

The Squeeze Theorem says:

and

imply

• The middle function can be difficult or oscillatory.
• The bounding functions must approach the same limit.
• Absolute-value bounds are especially useful.
• A bounded factor multiplied by a factor tending to zero often gives a limit of zero.
• The theorem works for two-sided, one-sided, and infinite-input limits.
• The fundamental limit \(\lim_{x\to0}\sin x/x=1\) is a major application.

The central strategy is: find an envelope that traps the difficult function and collapses to one common value.

Where We Go Next

The Squeeze Theorem is a comparison tool for determining exact limiting behavior. A broader question asks how functions compare when inputs become very large or approach critical values, even when no finite limit is the main focus.

The next lesson, Asymptotic Behavior, studies dominant terms, relative growth, asymptotes, and the large-scale behavior of functions.