Determining Limits Using the Squeeze Theorem

Introduction: Why a “trap” can reveal a limit 🌟

students, in calculus we often want to know what happens to a function near a point, even when the function itself is hard to evaluate directly. The Squeeze Theorem is a powerful tool for finding limits when a function is trapped between two other functions that both approach the same value. In that case, the middle function must approach that same value too.

Learning objectives

• Explain the main ideas and terminology behind the Squeeze Theorem.
• Apply AP Calculus BC reasoning to determine limits using the Squeeze Theorem.
• Connect the Squeeze Theorem to the broader topic of limits and continuity.
• Summarize how the Squeeze Theorem fits within Limits and Continuity.
• Use evidence and examples to justify a limit with the Squeeze Theorem.

This theorem is especially useful when a function includes oscillation, absolute values, trigonometric expressions, or other features that make direct substitution difficult. A classic example is $x^2\sin\left(\frac{1}{x}\right)$ near $x=0$. The sine part wiggles wildly, but the whole expression can still have a limit because the wiggles get squeezed into place.

The main idea of the Squeeze Theorem

The Squeeze Theorem says that if one function is always at or below a second function, and a third function is always at or above that second function, then the middle function shares the same limit as long as the top and bottom functions have the same limit.

In symbols, if for all $x$ near $a$,

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

and

$$\lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L,$$

then

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

This is also called the Sandwich Theorem or Pinching Theorem. The language matters because AP Calculus often asks students to state the conditions clearly: you need inequalities that hold near the point, and you need both bounding functions to approach the same number.

A helpful image is a narrow hallway. If students has two walls closing in on both sides and those walls meet at the same destination, then the path between them has to end at that same destination too. The function does not need to behave nicely by itself; it only needs to stay trapped between two better-behaved functions.

How to recognize when to use it

The Squeeze Theorem is useful when direct limit laws do not easily work because the function contains a part that oscillates or is hard to simplify. Common clues include:

$\sin$$\left($$\frac{1}{x}$$\right)$$ or $

$\cos$$\left($$\frac{1}{x}$$\right)$$ near $x=0

• Products like $x\sin\left(\frac{1}{x}\right)$ or $x^2\cos\left(\frac{1}{x}\right)$
• Expressions with absolute values, such as $x\,|\sin x|$
• Functions that are bounded by a simpler expression times a factor that goes to $0$

A key fact is that trigonometric functions like $\sin x$ and $\cos x$ are bounded. In particular,

$$-1 \le \sin x \le 1$$

and

$$-1 \le \cos x \le 1$$

for every real $x$. These inequalities are the starting point for many squeeze arguments.

For example, if we want to study $x\sin\left(\frac{1}{x}\right)$ as $x\to 0$, we know

$$-1 \le \sin\left(\frac{1}{x}\right) \le 1.$$

Multiplying by $x$ requires care because $x$ can be positive or negative, so a cleaner step is to use absolute values:

$$\left|x\sin\left(\frac{1}{x}\right)\right| \le |x|.$$

Since $|x|\to 0$ as $x\to 0$, the middle expression must also go to $0$.

Example 1: A classic oscillating limit

Let’s find

$$\lim_{x\to 0} x\sin\left(\frac{1}{x}\right).$$

Because $-1 \le \sin\left(\frac{1}{x}\right) \le 1$, multiplying by $|x|$ gives

$$-|x| \le x\sin\left(\frac{1}{x}\right) \le |x|.$$

Now consider the bounding functions:

$$\lim_{x\to 0}(-|x|)=0$$

and

$$\lim_{x\to 0}|x|=0.$$

Both bounds approach $0$, so by the Squeeze Theorem,

$$\lim_{x\to 0} x\sin\left(\frac{1}{x}\right)=0.$$

This example shows a major AP idea: a function can oscillate rapidly and still have a limit if the oscillation is controlled by a factor that shrinks to $0$.

A useful comparison is to think of a jumpy ball being held between two closing hands. The ball may bounce around, but if the hands get closer together and meet at the same point, the ball cannot end anywhere else.

Example 2: A trigonometric limit with a power factor

Now find

$$\lim_{x\to 0} x^2\cos\left(\frac{1}{x}\right).$$

Since $-1 \le \cos\left(\frac{1}{x}\right) \le 1$, multiplying by the nonnegative number $x^2$ gives

$$-x^2 \le x^2\cos\left(\frac{1}{x}\right) \le x^2.$$

Next, evaluate the bounds:

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

and

$$\lim_{x\to 0}x^2=0.$$

Therefore,

$$\lim_{x\to 0} x^2\cos\left(\frac{1}{x}\right)=0.$$

This is a common AP pattern: a bounded oscillating function times a factor that goes to $0$. The bounded part cannot prevent the whole product from approaching $0$.

Example 3: Using a graph or inequality reasoning

Suppose a function $f(x)$ satisfies

$$x^2\le f(x)\le x^2+x^4$$

for values of $x$ near $0$. To determine $\lim_{x\to 0} f(x)$, first find the limits of the bounds:

$$\lim_{x\to 0}x^2=0$$

and

$$\lim_{x\to 0}(x^2+x^4)=0.$$

Because both bounds approach $0$, the Squeeze Theorem gives

$$\lim_{x\to 0} f(x)=0.$$

This example is important because the middle function does not need a simple formula. In AP problems, the function may be defined piecewise, described graphically, or given through inequalities.

When studying continuity, this matters because if a function is not easy to evaluate directly, the Squeeze Theorem can still help show that the limit exists. Then, if $f(a)$ equals that limit, the function is continuous at $x=a$.

Conditions and common mistakes

To use the Squeeze Theorem correctly, students should check three things:

1. The target function is trapped between two other functions near the point.
2. The lower and upper functions have the same limit at the point.
3. The inequalities hold in some interval around the point, not just at one isolated value.

Common mistakes include:

• Forgetting to verify both bounds have the same limit.
• Using the theorem when the inequalities do not hold near the point.
• Trying to squeeze a function between bounds that do not actually trap it.
• Ignoring that multiplying inequalities by a negative number reverses the inequality signs.

A strong AP Calculus BC solution should clearly explain the bounding step and then explicitly name the theorem used. A good written justification might say, “Since $- |x| \le x\sin\left(\frac{1}{x}\right) \le |x|$ and both $- |x|$ and $|x|$ have limit $0$ as $x\to 0$, the Squeeze Theorem implies the limit is $0$.”

Connection to limits, continuity, and infinite behavior

The Squeeze Theorem fits into the broader unit on Limits and Continuity because it gives another way to prove a limit exists. This helps in several ways:

• It supports one-sided and two-sided limit thinking.
• It works with bounded oscillations that are difficult to handle with algebra alone.
• It can help establish continuity when the value at a point matches the limit.
• It connects to limits at infinity when a function is trapped between expressions that approach the same horizontal asymptote.

For example, if $f(x)$ is squeezed between two expressions that both approach $0$ as $x\to \infty$, then $f(x)$ also approaches $0$. That is a limit at infinity idea, which connects to end behavior and asymptotes.

The theorem also supports later calculus ideas. In derivative problems, certain expressions become easier when rewritten using a squeeze argument. In integral and series contexts, bounded behavior often plays a similar role, even if the formal theorem changes.

Conclusion

The Squeeze Theorem is one of the most useful ideas in AP Calculus BC because it turns a hard limit into an easier comparison problem. If students can find two functions that trap a third function and both bounds approach the same value, then the middle function must approach that value too. This theorem is especially valuable for limits involving trigonometric oscillation, absolute value, and small factors like $x$ or $x^2$ that force the expression toward a single answer. It is a key part of understanding how limits describe behavior near a point and how continuity depends on matching a limit with a function value.

Study Notes

• The Squeeze Theorem says that if $g(x)\le f(x)\le h(x)$ near $a$ and both $\lim_{x\to a}g(x)$ and $\lim_{x\to a}h(x)$ equal $L$, then $\lim_{x\to a}f(x)=L$.
• It is also called the Sandwich Theorem or Pinching Theorem.
• It is especially useful for expressions with $\sin\left(\frac{1}{x}\right)$, $\cos\left(\frac{1}{x}\right)$, or other bounded oscillations.
• A common strategy is to use $-1\le \sin x\le 1$ and $-1\le \cos x\le 1$.
• Absolute value can help create clear bounds, such as $\left|x\sin\left(\frac{1}{x}\right)\right|\le |x|$.
• The bounding functions must approach the same limit.
• The inequalities must hold near the point, not just at one value.
• The theorem is useful for proving limits, checking continuity, and understanding end behavior.
• On AP problems, always show the bounds first, then name the Squeeze Theorem explicitly.
• If a function is trapped between two expressions that both go to $0$, the function also goes to $0$.