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The Squeeze Theorem in calculus is a fundamental concept used to find limits of functions that are difficult to compute directly. It involves bounding a function between two others that converge to the same limit, ensuring the original function also converges to that limit. This theorem is particularly useful for functions with oscillatory behavior or discontinuities, such as those involving trigonometric expressions bounded by polynomials.

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## Definition and Importance of the Squeeze Theorem

### Overview of the Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a fundamental principle in calculus used to determine limits for functions that are not easily solvable by direct computation

### Purpose of the Squeeze Theorem

Facilitating the determination of limits for challenging functions

The Squeeze Theorem is invaluable when examining functions with oscillatory behavior or points of discontinuity

Strategic tool for evaluating limits

The Squeeze Theorem is a useful tool when other methods of limit evaluation are ineffective

### Proof of the Squeeze Theorem

The Squeeze Theorem is rigorously proven by showing that for any given positive number, there exists a corresponding value such that the bounding functions are within that value of the limit

## Application of the Squeeze Theorem

### Identifying bounding functions

To apply the Squeeze Theorem, two functions must be identified that serve as bounds for the function of interest and converge to the same limit

### Examples of using the Squeeze Theorem

Evaluating limits of functions with trigonometric expressions

The Squeeze Theorem can be used to evaluate limits of functions with trigonometric expressions bounded by polynomials

Correcting previous limit evaluations

The Squeeze Theorem can be used to correct previous limit evaluations, such as in the case of a function with a trigonometric expression in the numerator and a polynomial in the denominator

### Effectively applying the Squeeze Theorem

To effectively apply the Squeeze Theorem, a two-sided inequality must be established, the bounding functions must converge to a common limit, and the theorem must be invoked to determine the limit of the function

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