Squeeze Theorem

The Squeeze Theorem, also called the Sandwich Theorem or the Pinching Theorem, is a key result in calculus used to evaluate limits of functions that are difficult to analyze directly. If a function is bounded between two others that approach the same limit, it must converge to that limit as well. Though we have different theorems and formulas to find the limits of various types of functions, we always try to find the simpler way of solving them so that we get the solution quickly. The Squeeze Theorem is a fundamental tool for handling indeterminate forms and complex behaviors near a point.

Squeze Theorem Note

Use the Squeeze Theorem for oscillating or bounded functions trapped between two other functions whose limits match.

Use L’Hôpital’s Rule for fractions with indeterminate forms like 0/0 or ∞/∞ using derivatives.

Statement: Let f, g and h be real functions such that f(x) ≤ g(x) ≤ h(x) for all x in the common domain of definition. For some real number a, if

$\lim_{x\rightarrow a} f\left( x \right) \,\,=\,\,l\,\,=\,\,\lim_{x\rightarrow a} h\left( x \right)$

then

$\lim_{x \to a} g\left( x \right) \,\,=\,\,l$

The Squeeze Theorem is used to find the limit of a tricky function when direct substitution or standard algebra fails, typically by trapping it between two simpler functions that share the same limit.

How the Theorem Works

Setup

Imagine three functions—let’s call them f(x), g(x), and h(x)—such that f(x) ≤ g(x) ≤ h(x) when x is close to a number a.

$y=x^2$

$y=-x^2$

$g\left( x \right) =x^2\cos \left( \frac{1}{x} \right)$

Trap

Function g(x) is in the middle, “squeezed” below by f(x) and above by h(x).

Limits

If you take the limit as x approaches a for both outer functions and they equal the exact same number L:

$\lim_{x\rightarrow a} f\left( x \right) \,\,=\,L\,\,and\,\,\lim_{x\rightarrow a} h\left( x \right) \,\,=\,\,L$

Conclusion

Then the limit of the middle function as x approaches a must also be L:

$\lim_{x\rightarrow a} g\left( x \right) \,\,=\,\,L$

When to Use the Squeeze Theorem

Trigonometric Limits With Fast Oscillation

Use it when a function contains terms like sin(1/x) or cos(1/x) as x approaches 0 or infinity, where the trig value oscillates wildly and has no direct limit.

Standard Proofs

Use it solve fundamental limits, such as

$\lim_{x\rightarrow \infty} \frac{\sin \left( x \right)}{x}$

Algebraic Failure

Use it when normal methods like such as factoring, trigonometry substitutions, rationalization, or other algebraic manipulations.

Squeze Theorem Notes

It is a good idea to be familiar with radians and the unit circle. Many Squeeze Theorem problems will use trigonometric functions.

When in doubt, graph it! Is f(x) sandwiched between g(x) and h(x) on the necessary interval? Do their limits look the same at the right particular point? Graphing can help validate your answer.

Remember, the limits of your lower and upper bounds must match. Otherwise the theorem can’t be used.

See the video Most Calculus Students Use L’Hôpital’s Rule Here… But They Shouldn’t.

Requirements to Apply It

Find Bounds

You must find a lower function f(x) and an upper function h(x) such that f(x) ≤ g(x) ≤ h(x) for all x near your target point.

Match Limits

The limits of the outer functions must approach the exact same finite value L as x approaches a:

$\lim_{x\rightarrow a} f\left( x \right) \,\,=\,\,\lim_{x\rightarrow a} h\left( x \right) \,\,=\,\,L$

Conclusion

The middle function’s limit is forced to be L as well.

Using the Squeeze Theorem

See Understanding the Squeeze Theorem, Squeeze Theorem Examples for an explanation and examples.

**Sandwich (Squeeze) Theorem:** When a function is trapped between two functions that approach the same limit, it must approach that limit too. A simple idea with powerful applications in calculus. – degamma_maths

Why not use L’Hôpital’s Rule?

Start with the following limit.

$\lim_{x\rightarrow 0} \frac{\sin \left( x \right)}{x}$

There are two main methods: L’Hôpital’s Rule (algebraic check) or the Squeeze Theorem (geometric proof).

We can use the theorem to find tricky limits like sin(x)/x at x = 0, by “squeezing” sin(x)/x between two nicer functions and using them to find the limit at x = 0.

Step 1: Identify the Indeterminate Form

Directly substituting x = 0 into the expression gives: sin(0)/0 = 0/0

This is an indeterminate form, meaning you must apply a limit evaluation technique.

Step 2: Apply L’Hôpital’s Rule

Because the expression yields 0/0, you can differentiate the numerator and the denominator separately with respect to x:

Derivative of the numerator: d/dx sin(x) = cos(x)
Derivative of the denominator: d/dx x = 1

Now, substitute these derivatives back into the limit:

$\lim_{x\rightarrow 0} \frac{\sin \left( x \right)}{x}\,\,=\,\,\lim_{x\rightarrow 0} \frac{\cos \left( x \right)}{1}$

Step 3: Evaluate by Direct Substitution

Since cos(x) is continuous at x = 0, substitute 0 directly into the simplified expression:

$\lim_{x\rightarrow 0} \cos \left( x \right) \,\,=\,\,\cos \left( 0 \right) \,\,=\,\,1$

Step 4: Answer

The limit as x approaches 0 of sin(x)/x is exactly 1.

References

CharmingCantalope. “Can Someone Help Explain the Squeeze Theorem? : R/Calculus.” reddit. Accessed July 27, 2026. https://www.reddit.com/r/calculus/comments/gi4ukg/can_someone_help_explain_the_squeeze_theorem/?solution=54a3c93c77dff6e854a3c93c77dff6e8.

A way I personally recognize the use of the squeeze theorem is when a trig function is involved. Now, you use the squeeze theorem is you have your “base” or given function and then you find two functions which are greater than and less than or equal to the given function. You then take the limit of these two found functions and, so long as they are equal to each other in the end, that limit will be the limit of your given function (by the squeeze theorem). An example would be if you had the limit as x approaches 0 of x^2*sin(1/x). Notice, 1/x will approach infinity as x approaches zero, and the limit as x approaches infinity of sinx does not exist. So, we cannot take the limit directly, but notice that sin(1/x) will be at most 1 and at least -1. Therefore -1<=sin(1/x)<=1. This looks a lot like the original function but we need to times everything by x² to make it the original, so the inequality becomes -x^{2<=x2*sin(1/x)<=x2.} Now, the limit as x approaches zero of x² is really easy (it’s just zero), so taking the limit of the whole inequality would turn into 0<= the limit as x approaches 0 of x^2*sin(1/x)<= 0. Now, since both of the functions we derived are equal to each other and the given function is between them, by the squeeze theorem the original limit is equal to zero. If you noticed at the beginning of the example, we use the squeeze theorem to evaluate limits which would otherwise be impossible to evaluate.

“L’Hopital’s Rule.” CalcWorkshop, September 6, 2015. https://calcworkshop.com/derivatives/lhopitals-rule/.

“L’Hôpital’s Rule.” fiveable, n.d. Accessed July 29, 2026. https://fiveable.me/calc-i/unit-4/8-lhopitals-rule/study-guide/9iaCfSpJUNzlpEyh.

[ ℰ ] Martin, Isaac. “Squeeze Theorem and the Limit of sin(x)/x.” web.ma.utexas.edu, October 14, 2025. https://web.ma.utexas.edu/users/ikmartin/pages/writing/blog/calc_squeeze/.

McBride, Martin. “GraphicMaths.” Limits That Fail L’Hôpital’s Rule, August 28, 2025. https://graphicmaths.com/pure/differentiation/lhopitals-rule-fails/.

We can often use L’Hopital’s rule to find the limit of an indeterminate form. In many cases, L’Hôpital’s rule is the only way to find a solution. But we should bear in mind that it is not the only method available, and might not be the best method in every situation. In particular, just because L’Hospital’s rule fails to find a limit, that does not necessarily mean that no limit exists. In this article, we will look at a case where L’Hôpital’s rule doesn’t work. We will see two alternative ways to find the limit.

[ ℰ ] McLean, Rachel. “Understanding the Squeeze Theorem.” Outlier, February 21, 2023. https://articles.outlier.org/what-is-the-squeeze-theorem.

Miglani, Komal. “Sandwich Theorem.” Careers360, September 9, 2024. https://www.careers360.com/maths/sandwich-theorem-topic-pge.

“Sandwich Theorem in Calculus Explained Clearly.” Vedantu, October 25, 2022. https://www.vedantu.com/maths/sandwich-theorem.

Simpson, Roy. “1.3: Limit Laws and the Squeeze Theorem.” LibreTexts, October 23, 2022. https://math.libretexts.org/Courses/Cosumnes_River_College/Math_400%3A_Calculus_I_-_Differential_Calculus/01%3A_Learning_Limits/1.03%3A_Limit_Laws_and_the_Squeeze_Theorem.

“Squeeze Theorem.” CalcWorkshop, October 16, 2015. https://calcworkshop.com/limits/squeeze-theorem/.

“Squeeze Theorem.” Wikipedia, July 7, 2026. https://en.wikipedia.org/wiki/Squeeze_theorem.

“Squeeze Theorem – Formula, Proof, Examples.” CUEMATH, Accessed July 27, 2026. https://www.cuemath.com/calculus/squeeze-theorem/.

“The Squeeze Theorem.” 3.6 The Squeeze Theorem. SFU, Accessed July 27, 2026. https://www.sfu.ca/math-coursenotes/Math%20157%20Course%20Notes/sec_TrigLimits.html.

[ ℰ ] “The Squeeze Theorem.” Calculus I. lumen, Accessed July 27, 2026. https://courses.lumenlearning.com/calculus1/chapter/the-squeeze-theorem/.

Videos

How to Use the Squeeze Theorem for Limits | Calculus 1 & AP Exam

In this video, you’ll learn how to use the Squeeze Theorem to evaluate limits in Calculus 1 and AP Calculus. I walk through step-by-step examples for limits as x approaches a value and as x goes to infinity, so you know exactly when and how to apply the theorem on exams.

The Squeeze Theorem of Calculus Explained

This tutorial delves into the Squeeze Theorem, a fundamental concept in calculus that offers a powerful tool for evaluating limits, especially when direct substitution is not possible. Aimed at students, educators, or anyone looking to enhance their understanding of mathematical analysis, this video simplifies the theorem, making it accessible and easy to grasp for viewers at all levels.

The Squeeze Theorem, also known as the Sandwich Theorem or Pinching Theorem, is based on the principle that if a function is bounded by two other functions that have the same limit at a certain point, then the function of interest must also converge to that same limit.

Through detailed examples, the tutorial demonstrates how to apply the Squeeze Theorem to solve limit problems involving trigonometric, polynomial, and other types of functions. Viewers will learn how to identify situations where the theorem can be effectively used and see step-by-step how to set up the bounding functions.

Additionally, the video highlights common pitfalls in applying the Squeeze Theorem and offers tips for avoiding these mistakes. By the end of this tutorial, viewers will have a solid grasp of using the Squeeze Theorem to find limits, enhancing their problem-solving skills in calculus. Join us to unlock the potential of this essential analytical tool.

Squeeze theorem or sandwich theorem | Limits | Differential Calculus | Khan Academy

Squeeze Theorem

This calculus video tutorial explains the squeeze theorem with trig functions like sin and cos (1/x). It explains the definition of the theorem and how to evaluate functions and limits with inequalities.

L’hôpital’s Rule & The squeeze Theorem

Most Calculus Students Use L’Hôpital’s Rule Here… But They Shouldn’t

Do you automatically reach for L’Hôpital’s Rule every time you see a difficult limit? If so, you’re not alone—but you might be making one of the most common mistakes in calculus. In this video, we’ll solve a classic limit that looks like it requires L’Hôpital’s Rule but actually should be solved using the Squeeze Theorem. Then we’ll review the only two indeterminate forms where L’Hôpital’s Rule applies: 0/0 and ∞/∞, with examples of each. If you’re studying for Calculus 1, Calculus AB/BC, or preparing for an exam, this is a concept you absolutely need to master.

[ ℰ ] This exceptional reference is highly recommended for your consideration.