Calculus I · Limits and Continuity · lesson

The Squeeze Theorem

Learn The Squeeze Theorem with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.

Course progress19 of 47 core pages19 of 47

Where this chapter fits

Chapter 3: Trigonometric limits

Use squeezing, the fundamental sine limit, identities, and scaling to make trigonometric behavior predictable.

Reading lens: Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for? Keep that question in view while reading The Squeeze Theorem; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects Finite Limit Decision Tree to Why sin(x)/x Approaches 1. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Use upper and lower bounds to determine a limit even when the middle function oscillates or is difficult to evaluate directly.

The Squeeze Theorem and Trigonometric Limits

When Direct Algebra Is Not Enough

Some functions refuse to simplify into a friendly formula. The Squeeze Theorem handles them by trapping their outputs between two simpler functions.

Concept

Suppose three people walk through a doorway side by side. The person in the middle cannot end up above the person on the top or below the person on the bottom. If the top and bottom people are forced toward the same height, the middle person is forced there too. That is the Squeeze Theorem.

Theorem

Squeeze Theorem

Suppose that for all $x$ sufficiently close to $a$ ,

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

and suppose

\lim_{x\to a}g(x)=L \qquad\text{and}\qquad \lim_{x\to a}h(x)=L.

Then

\boxed{\lim_{x\to a}f(x)=L}.

Guided walkthrough

A number trapped between shrinking walls

Suppose $f(x)$ satisfies

-x^2\le f(x)\le x^2

near $x=0$ . Find $\lim_{x\to0}f(x)$ .

Show worked solution

The lower wall approaches

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

The upper wall approaches

\lim_{x\to0}x^2=0.

Because $f(x)$ is trapped between two functions that both approach $0$ ,

\boxed{\lim_{x\to0}f(x)=0}.

We do not need a formula for $f$ . The trap is enough.

Interactive checksqueeze-flow-01

Evaluate $\lim_{x\to0}x^2\sin(1/x)$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Use $|\sin(1/x)|\le1$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Bounded oscillation times a shrinking factor

The inequalities

-1\le\sin u\le1

and

-1\le\cos u\le1

hold for every real $u$ . If a bounded trigonometric factor is multiplied by something that approaches zero, the product is often squeezed to zero.

Worked example

An oscillating function forced to zero

Evaluate

\lim_{x\to0}x^2\sin\left(\frac1x\right).

Show worked solution

Start with the universal sine bound:

-1\le\sin\left(\frac1x\right)\le1.

Since $x^2\ge0$ , multiply every part by $x^2$ without reversing the inequalities:

-x^2\le x^2\sin\left(\frac1x\right)\le x^2.

Now take limits of the outer functions:

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

Therefore,

\boxed{\lim_{x\to0}x^2\sin\left(\frac1x\right)=0}.

The sine factor continues oscillating. The shrinking factor $x^2$ reduces the size of every oscillation until the graph is trapped near zero.

Graph reading guide

The oscillating function is trapped between $-x^2$ and $x^2$ , both of which approach zero.

Interactive graph exploration

Functions: $y=x^2\sin(1/x)$ , $y=x^2$ , and $y=-x^2$ . Window: $-0.35\le x\le0.35$ , $-0.13\le y\le0.13$ . Use at least 1500 samples on each side of zero for the oscillating curve. Do not connect across $x=0$ , where the displayed formula is undefined.

Worked example

Absolute-value squeeze

Evaluate

\lim_{x\to0}x\cos\left(\frac{5}{x}\right).

Show worked solution

Because $|\cos u|\le1$ ,

\left|x\cos\left(\frac5x\right)\right| \le |x|.

This means

-|x|\le x\cos\left(\frac5x\right)\le|x|.

Both outer functions approach zero, so

\boxed{\lim_{x\to0}x\cos\left(\frac5x\right)=0}.

Problem

When a function contains $\sin(1/x)$ , $\cos(1/x)$ , or another bounded oscillating factor, ask whether the rest of the expression approaches zero. The inequality

|\text{bounded factor}\times\text{shrinking factor}| \le C|\text{shrinking factor}|

is often the entire argument.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.

Calculus I · Limits and Continuity · lesson

The Squeeze Theorem

Learn The Squeeze Theorem with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.

Course progress19 of 47 core pages19 of 47

Where this chapter fits

Chapter 3: Trigonometric limits

Use squeezing, the fundamental sine limit, identities, and scaling to make trigonometric behavior predictable.

This page connects Finite Limit Decision Tree to Why sin(x)/x Approaches 1. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Use upper and lower bounds to determine a limit even when the middle function oscillates or is difficult to evaluate directly.

The Squeeze Theorem and Trigonometric Limits

When Direct Algebra Is Not Enough

Some functions refuse to simplify into a friendly formula. The Squeeze Theorem handles them by trapping their outputs between two simpler functions.

Concept

Theorem

Squeeze Theorem

Suppose that for all $x$ sufficiently close to $a$ ,

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

and suppose

\lim_{x\to a}g(x)=L \qquad\text{and}\qquad \lim_{x\to a}h(x)=L.

Then

\boxed{\lim_{x\to a}f(x)=L}.

Guided walkthrough

A number trapped between shrinking walls

Suppose $f(x)$ satisfies

-x^2\le f(x)\le x^2

near $x=0$ . Find $\lim_{x\to0}f(x)$ .

Show worked solution

The lower wall approaches

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

The upper wall approaches

\lim_{x\to0}x^2=0.

Because $f(x)$ is trapped between two functions that both approach $0$ ,

\boxed{\lim_{x\to0}f(x)=0}.

We do not need a formula for $f$ . The trap is enough.

Interactive checksqueeze-flow-01

Evaluate $\lim_{x\to0}x^2\sin(1/x)$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Use $|\sin(1/x)|\le1$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Bounded oscillation times a shrinking factor

The inequalities

-1\le\sin u\le1

and

-1\le\cos u\le1

hold for every real $u$ . If a bounded trigonometric factor is multiplied by something that approaches zero, the product is often squeezed to zero.

Worked example

An oscillating function forced to zero

Evaluate

\lim_{x\to0}x^2\sin\left(\frac1x\right).

Show worked solution

Start with the universal sine bound:

-1\le\sin\left(\frac1x\right)\le1.

Since $x^2\ge0$ , multiply every part by $x^2$ without reversing the inequalities:

-x^2\le x^2\sin\left(\frac1x\right)\le x^2.

Now take limits of the outer functions:

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

Therefore,

\boxed{\lim_{x\to0}x^2\sin\left(\frac1x\right)=0}.

The sine factor continues oscillating. The shrinking factor $x^2$ reduces the size of every oscillation until the graph is trapped near zero.

Graph reading guide

The oscillating function is trapped between $-x^2$ and $x^2$ , both of which approach zero.

Interactive graph exploration

Worked example

Absolute-value squeeze

Evaluate

\lim_{x\to0}x\cos\left(\frac{5}{x}\right).

Show worked solution

Because $|\cos u|\le1$ ,

\left|x\cos\left(\frac5x\right)\right| \le |x|.

This means

-|x|\le x\cos\left(\frac5x\right)\le|x|.

Both outer functions approach zero, so

\boxed{\lim_{x\to0}x\cos\left(\frac5x\right)=0}.

Problem

When a function contains $\sin(1/x)$ , $\cos(1/x)$ , or another bounded oscillating factor, ask whether the rest of the expression approaches zero. The inequality

|\text{bounded factor}\times\text{shrinking factor}| \le C|\text{shrinking factor}|

is often the entire argument.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

Chapter 3: Trigonometric limits

The Squeeze Theorem and Trigonometric Limits

When Direct Algebra Is Not Enough

Bounded oscillation times a shrinking factor

Related resources

Original instruction with traceable references.

Chapter 3: Trigonometric limits

The Squeeze Theorem and Trigonometric Limits

When Direct Algebra Is Not Enough

Bounded oscillation times a shrinking factor

Related resources

Original instruction with traceable references.