Squeeze Theorem

Definition

The squeeze theorem, also known as the sandwich theorem or the pinching theorem, is a fundamental concept in calculus that allows us to determine the limit of a function when it is trapped or 'squeezed' between two other functions whose limits are known.

5 Must Know Facts For Your Next Test

1. The squeeze theorem states that if a function $f(x)$ is bounded above by a function $g(x)$ and bounded below by a function $h(x)$, and if the limits of $g(x)$ and $h(x)$ are equal as $x$ approaches a particular value, then the limit of $f(x)$ is also equal to that value.
2. The squeeze theorem is particularly useful when dealing with functions that are difficult to evaluate directly, but can be bounded above and below by simpler functions.
3. The squeeze theorem can be used to establish the limits of trigonometric functions, such as $rac{ ext{sin}(x)}{x}$ and $rac{ ext{cos}(x) - 1}{x}$ as $x$ approaches 0.
4. The squeeze theorem can also be used to determine the limits of rational functions, exponential functions, and other types of functions.
5. The squeeze theorem is a powerful tool for analyzing the behavior of functions and establishing their limits, which is a crucial concept in calculus.

Review Questions

• Explain the key idea behind the squeeze theorem and how it can be used to determine the limit of a function.
  • The squeeze theorem states that if a function $f(x)$ is bounded above by a function $g(x)$ and bounded below by a function $h(x)$, and if the limits of $g(x)$ and $h(x)$ are equal as $x$ approaches a particular value, then the limit of $f(x)$ is also equal to that value. This is because the function $f(x)$ is 'squeezed' between the two bounding functions, and as the limits of the bounding functions converge, the limit of $f(x)$ must also converge to the same value. The squeeze theorem is particularly useful when dealing with functions that are difficult to evaluate directly, but can be bounded above and below by simpler functions.
• Describe how the squeeze theorem can be used to establish the limits of trigonometric functions, such as $rac{ ext{sin}(x)}{x}$ and $rac{ ext{cos}(x) - 1}{x}$ as $x$ approaches 0.
  • The squeeze theorem can be used to establish the limits of trigonometric functions, such as $rac{ ext{sin}(x)}{x}$ and $rac{ ext{cos}(x) - 1}{x}$ as $x$ approaches 0. For example, to find the limit of $rac{ ext{sin}(x)}{x}$ as $x$ approaches 0, we can use the fact that $0 ext{ } extless ext{ } ext{sin}(x) ext{ } extless ext{ } x$ for all $x$ near 0. This means that $0 ext{ } extless ext{ } rac{ ext{sin}(x)}{x} ext{ } extless ext{ } 1$. Since the limits of the bounding functions, 0 and 1, are both equal to 1 as $x$ approaches 0, the squeeze theorem tells us that the limit of $rac{ ext{sin}(x)}{x}$ must also be 1. Similarly, the squeeze theorem can be used to show that the limit of $rac{ ext{cos}(x) - 1}{x}$ as $x$ approaches 0 is 0.
• Analyze how the squeeze theorem can be applied to determine the limits of other types of functions, such as rational functions and exponential functions.
  • The squeeze theorem can be applied to determine the limits of a wide variety of functions, including rational functions and exponential functions. For rational functions, the squeeze theorem can be used to establish the limits of expressions like $rac{x^2 - 1}{x - 1}$ as $x$ approaches 1, by bounding the function above and below by simpler rational expressions with known limits. Similarly, for exponential functions, the squeeze theorem can be used to determine the limits of expressions like $rac{e^x - 1}{x}$ as $x$ approaches 0, by bounding the function above and below by simpler exponential expressions. The key idea is to identify appropriate bounding functions that are easier to analyze, and then use the squeeze theorem to conclude that the limit of the original function must be the same as the limits of the bounding functions. This makes the squeeze theorem a powerful and versatile tool for analyzing the behavior of a wide range of functions in calculus.

Related terms

Limit: The value that a function approaches as the input variable gets closer to a particular point.

Continuity: A function is continuous at a point if it has a well-defined value at that point and the function's value can be made arbitrarily close to the function's value at that point by making the input variable sufficiently close to the point.

Bounded Function: A function is bounded if there exists some positive number M such that the absolute value of the function is less than or equal to M for all values of the input variable.