Key takeaway: The squeeze theorem says that if a function is trapped between two functions that approach the same value L, then the trapped function must also approach L.
Grade 12 – AP Calculus ABAP Calculus AB Unit 1Calculus
Definition of Squeeze Theorem
The squeeze theorem handles functions that are too messy to analyze directly. If you cannot find where \(f\) is heading, trap it between a lower function \(g\) and an upper function \(h\) whose behavior you do know. When both walls close in on the same number, \(f\) has nowhere else to go.
Other names are the sandwich theorem and the pinching theorem. It is the standard way to deal with oscillating factors such as sine and cosine, whose size is always limited to between -1 and 1. The inequality only needs to hold near the point of interest, not on the whole domain.
Formula and notation
Suppose that \(g(x)\le f(x)\le h(x)\) for all \(x\) near \(a\) (except possibly at \(a\)). Then
\[\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L \implies \lim_{x\to a}f(x)=L.\]
The same statement holds as \(x\to\infty\), with the inequalities valid for large \(x\).
Key facts
- Both bounding functions must have the same value; one alone is not enough.
- Bounds for oscillating terms come from -1 ≤ sin x ≤ 1 and -1 ≤ cos x ≤ 1.
- When you multiply an inequality by a negative number, reverse its direction.
- The method proves the value exists, with no need to know f exactly.
Squeeze Theorem example
Find \(\lim_{x\to\infty}\dfrac{\sin x}{x}\).
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Sine always lies between -1 and 1, so for positive \(x\), dividing by \(x\) gives \(-\frac{1}{x}\le\frac{\sin x}{x}\le\frac{1}{x}\).
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As \(x\) grows, \(\frac{1}{x}\) heads to 0, and so does \(-\frac{1}{x}\).
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Both walls approach 0, so the trapped function must approach the same number.
By the squeeze theorem, the value is 0.
Common mistake
Watch out
Kyra says \(\lim_{x\to\infty}\frac{\sin x}{x}\) does not exist because \(\sin x\) keeps oscillating forever.
Why it is wrong
The numerator oscillates but stays between -1 and 1, while the denominator grows without bound. The oscillation is squeezed flat, as the pinched curve above shows, and the outputs go to 0.
Related terms and lessons
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