How to Use Squeeze Theorem (and when!) (AP)

1) What You Need to Know

What it is (and why you care)

The Squeeze Theorem (aka Sandwich Theorem) is your go-to tool for limits when:

• the expression oscillates (usually trig like $\sin(1/x)$, $\cos(1/x)$),
• the expression is messy but you can trap it between two simpler functions,
• or you need to prove a standard limit (especially trig limits) from basic inequalities.

On AP Calc BC, it shows up most often in:

• limits as $x \to 0$ involving trig + powers of $x$,
• limits as $x \to \infty$ (or sequences $n \to \infty$) where something is bounded,
• justification questions (“explain using Squeeze Theorem”).

The theorem (precise statement)

If for all $x$ near $a$ (possibly excluding $a$ itself),

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

and

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

then

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

Same idea for sequences: if $a_n \le b_n \le c_n$ for all sufficiently large $n$ and $\lim a_n = \lim c_n = L$, then $\lim b_n = L$.

When you should use it

Use Squeeze Theorem when direct substitution fails and:

• you can find two functions that are easy to limit, and
• those bounds meet at the same limit.

Most common “Squeeze signals”:

• bounded trig: $-1 \le \sin(\cdot) \le 1$ and $-1 \le \cos(\cdot) \le 1$
• absolute value: $-|u| \le u \le |u|$ and if $|f(x)| \le g(x)$ with $g(x) \to 0$ then $f(x) \to 0$
• products with oscillation: something like $x^k \sin(1/x)$ or $\frac{\sin n}{n}$

Critical reminder: Squeeze only works if you show the inequality holds in a punctured neighborhood of $a$ (or for all sufficiently large $n$), and both bounds have the same limit.

2) Step-by-Step Breakdown

The “Sandwich” method (limits)

1) Identify the hard piece

• Usually the oscillating/unknown part: $\sin(1/x)$, $\cos(1/x)$, or something trapped by absolute value.

2) Write a true inequality that traps it

• For trig: $-1 \le \sin(\theta) \le 1$ and $-1 \le \cos(\theta) \le 1$ for any $\theta$.
• For absolute value: $-|u| \le u \le |u|$.

3) Multiply (or otherwise transform) the inequality carefully

• If you multiply by something that could be negative, the inequality direction can flip.
• A safe move: multiply by a nonnegative expression like $|x|^k$ or $x^2$.

4) Compute the limits of the two bounds

• If both go to the same $L$, you’ve squeezed.

5) Conclude the middle limit equals that same $L$

• State explicitly: “Since $g(x) \le f(x) \le h(x)$ and $\lim g = \lim h = L$, then $\lim f = L$.”

Mini worked walkthrough (classic oscillation)

Evaluate $\lim_{x \to 0} x^2 \sin(1/x).$

1) Use trig bound: $-1 \le \sin(1/x) \le 1$.

2) Multiply by $x^2$ (note $x^2 \ge 0$ so inequality direction stays):

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

3) Take limits as $x \to 0$:

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

4) Bounds match, so

$\lim_{x \to 0} x^2\sin(1/x) = 0.$

The “absolute value squeeze” (super common)

If you can show

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

and

$\lim_{x \to a} g(x) = 0,$

then automatically

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

Reason: $-|f(x)| \le f(x) \le |f(x)|$ and $0 \le |f(x)| \le g(x)$.

Step-by-step for sequences

To evaluate $\lim_{n \to \infty} b_n$ by squeeze:
1) Find $a_n$ and $c_n$ with $a_n \le b_n \le c_n$ for all sufficiently large $n$.
2) Compute $\lim a_n$ and $\lim c_n$.
3) If they match, conclude $\lim b_n$.

3) Key Formulas, Rules & Facts

Core Squeeze Theorem rules

Standard bounds you should know cold

Useful “squeeze-ready” patterns

Warning: Squeeze is not “the limit of the middle is between the limits.” The bounds’ limits must be the same.

4) Examples & Applications

Example 1: Oscillation + power of $x$

Evaluate $\lim_{x \to 0} x\cos(1/x).$

• Bound: $-1 \le \cos(1/x) \le 1$.
• Multiply by $|x|$ (safe, nonnegative):

$-|x| \le x\cos(1/x) \le |x|.$

• Limits: $\lim_{x \to 0} (-|x|)=0$ and $\lim_{x \to 0} |x|=0$.

So

$\lim_{x \to 0} x\cos(1/x)=0.$

Exam variation: They may give $x^3\sin(5/x)$ or $\sqrt{|x|}\cos(1/x)$. Same idea: trap trig between $-1$ and $1$, then the outside factor goes to $0$.

Example 2: Sequence squeeze

Evaluate $\lim_{n \to \infty} \frac{\sin(n^2)}{n}.$

• Since $|\sin(n^2)| \le 1$,

$\left|\frac{\sin(n^2)}{n}\right| \le \frac{1}{n}.$

• And $\lim_{n \to \infty} \frac{1}{n} = 0$.

Therefore

$\lim_{n \to \infty} \frac{\sin(n^2)}{n} = 0.$

Exam variation: could be $\frac{\cos(\sqrt{n})}{n^{1/3}}$ or $\frac{5\sin(n)}{n^2}$.

Example 3: Squeezing a “difference that goes to 0”

Evaluate $\lim_{x \to 0} \big(\cos(x) - 1\big)\sin(1/x).$

• Use $|\sin(1/x)| \le 1$:

$\left|\big(\cos(x) - 1\big)\sin(1/x)\right| \le |\cos(x)-1|.$

• As $x \to 0$, $\cos(x) \to 1$, so $|\cos(x)-1| \to 0$.

So by absolute value squeeze,

$\lim_{x \to 0} \big(\cos(x) - 1\big)\sin(1/x)=0.$

Exam variation: Replace $\cos(x)-1$ with any factor that goes to $0$ (like $x^2$, $\ln(1+x)$, $e^x-1$).

Example 4: The classic trig limit (why squeeze is famous)

Evaluate $\lim_{x \to 0} \frac{\sin x}{x}.$

A standard squeeze proof uses inequalities (often derived geometrically) that imply for $x$ near $0$:

$\cos x \le \frac{\sin x}{x} \le 1$

(for $x>0$; a similar argument handles $x<0$).

Then as $x \to 0$,

$\lim_{x \to 0} \cos x = 1, \qquad \lim_{x \to 0} 1 = 1,$

so

$\lim_{x \to 0} \frac{\sin x}{x} = 1.$

Why this matters: This limit is the gateway to derivatives of $\sin x$ and $\cos x$ and many trig limits.

5) Common Mistakes & Traps

1) “The bounds don’t meet”

• Wrong: You show $g(x) \le f(x) \le h(x)$ but $\lim g \ne \lim h$.
• Why wrong: Squeeze Theorem requires both bounding limits to be the same.
• Fix: Choose tighter bounds or a different technique (maybe algebra, L’Hôpital, or series—if allowed).

2) Multiplying by something negative (and not flipping inequalities)

• Wrong: Start with $-1 \le \sin(1/x) \le 1$ and multiply by $x$ (which changes sign).
• Why wrong: If $x<0$, inequality directions flip, breaking your squeeze.
• Fix: Multiply by $|x|$ or by $x^2$, or split into cases $x>0$ and $x<0$.

3) Assuming the inequality holds “at the limit point”

• Wrong: Only checking inequality at $x=a$ instead of near $a$.
• Why wrong: Limits care about values arbitrarily close to $a$.
• Fix: State “for $0<|x-a|<\delta$” (conceptually) or “for $x$ sufficiently close to $a$.”

4) Forgetting absolute value is your friend

• Wrong: Trying to bound $x\sin(1/x)$ without absolute values and getting tangled in signs.
• Why wrong: The cleanest squeeze is often $|\sin(\cdot)| \le 1$.
• Fix: Use $|x\sin(1/x)| \le |x|$ then go straight to $0$.

5) Using false trig bounds

• Wrong: Writing something like $\sin(\theta) \le \theta$ for all $\theta$ (not globally true), or using degree/radian confusion.
• Why wrong: Squeeze requires correct inequalities.
• Fix: Stick to always-true bounds: $-1 \le \sin(\theta) \le 1$, $-1 \le \cos(\theta) \le 1$, and known standard inequalities for $\sin x/x$ in radians.

6) Not stating the final squeeze conclusion clearly

• Wrong: You compute bounding limits but never explicitly conclude $\lim f = L$.
• Why wrong: AP graders like the logical chain.
• Fix: Write one clean sentence invoking the theorem.

7) Trying to squeeze when the function isn’t actually trapped

• Wrong: Picking bounds that are sometimes above/below but not consistently.
• Why wrong: You need $g(x) \le f(x) \le h(x)$ in an interval near the point.
• Fix: Use absolute values or restrict domain appropriately.

8) Mixing up “bounded” with “goes to 0”

• Wrong: Thinking “since $\sin(1/x)$ is bounded, its limit is $0$.”
• Why wrong: Bounded does not imply it approaches a value.
• Fix: You need a shrinking factor (like $x$, $x^2$, $1/n$) to force the product/ratio to $0$.

6) Memory Aids & Quick Tricks

7) Quick Review Checklist

• You can state Squeeze Theorem: $g \le f \le h$ and $\lim g = \lim h = L \Rightarrow \lim f = L$.
• You know the universal bounds: $-1 \le \sin(\theta),\cos(\theta) \le 1$ and $|\sin(\theta)|,|\cos(\theta)| \le 1$.
• You default to absolute values to avoid sign headaches: show $|f(x)| \le g(x)$.
• You only multiply inequalities by expressions you know are nonnegative (or you split cases).
• You verify the inequality holds near the limit point (or for all large $n$ in sequences).
• You check the two bounding limits are the same before concluding.
• You can do core AP patterns quickly: $x^k\sin(1/x) \to 0$ (for $k>0$) and $\frac{\sin n}{n} \to 0$.

One clean squeeze setup is often worth more than a page of algebra—aim for the tight, correct bounds and you’ll be fine.