Determining Limits Using the Squeeze Theorem

Introduction

Key Concepts

Understanding Limits and Continuity

In calculus, the concept of a limit is essential for analyzing the behavior of functions as they approach a particular point. Formally, the limit of a function \( f(x) \) as \( x \) approaches \( c \) is denoted as: $$ \lim_{{x \to c}} f(x) = L $$ This expression means that as \( x \) gets arbitrarily close to \( c \), \( f(x) \) approaches the value \( L \). Continuity at a point \( c \) implies that: $$ \lim_{{x \to c}} f(x) = f(c) $$ A function is continuous over an interval if it is continuous at every point within that interval. Understanding limits and continuity is crucial for solving calculus problems involving derivatives and integrals.

The Squeeze Theorem: Definition and Importance

The Squeeze Theorem, also known as the Sandwich Theorem, is a limit property used to find the limit of a function that is difficult to evaluate directly. The theorem states that if three functions \( f(x) \), \( g(x) \), and \( h(x) \) satisfy: $$ f(x) \leq g(x) \leq h(x) $$ for all \( x \) in an open interval around \( c \) (except possibly at \( c \) itself), and: $$ \lim_{{x \to c}} f(x) = \lim_{{x \to c}} h(x) = L $$ then: $$ \lim_{{x \to c}} g(x) = L $$ This theorem is particularly useful when \( g(x) \) is "squeezed" between \( f(x) \) and \( h(x) \) and directly finding its limit is complex.

Applying the Squeeze Theorem: Steps and Methodology

To apply the Squeeze Theorem effectively, follow these systematic steps:

1. Identify the Function: Determine the function \( g(x) \) whose limit you need to evaluate.
2. Find Bounding Functions: Identify two functions \( f(x) \) and \( h(x) \) such that \( f(x) \leq g(x) \leq h(x) \) holds in an open interval around the point of interest.
3. Evaluate the Limits of Bounding Functions: Compute \( \lim_{{x \to c}} f(x) \) and \( \lim_{{x \to c}} h(x) \).
4. Conclude the Limit of \( g(x) \): If both bounding functions approach the same limit \( L \), then \( \lim_{{x \to c}} g(x) = L \).

This methodological approach ensures that even if direct evaluation of \( g(x) \) is not straightforward, its behavior can be inferred through the bounding functions.

Examples Using the Squeeze Theorem

Example 1: Evaluating \( \lim_{{x \to 0}} x^2 \cos\left(\frac{1}{x}\right) \)

To find: $$ \lim_{{x \to 0}} x^2 \cos\left(\frac{1}{x}\right) $$ Step 1: Identify \( g(x) = x^2 \cos\left(\frac{1}{x}\right) \). Step 2: Find bounding functions. Since \( -1 \leq \cos\left(\frac{1}{x}\right) \leq 1 \), multiply all parts by \( x^2 \) (which is always non-negative): $$ -x^2 \leq x^2 \cos\left(\frac{1}{x}\right) \leq x^2 $$ Thus, \( f(x) = -x^2 \) and \( h(x) = x^2 \). Step 3: Evaluate the limits of the bounding functions: $$ \lim_{{x \to 0}} -x^2 = 0 \quad \text{and} \quad \lim_{{x \to 0}} x^2 = 0 $$ Step 4: By the Squeeze Theorem, $$ \lim_{{x \to 0}} x^2 \cos\left(\frac{1}{x}\right) = 0 $$

Example 2: Evaluating \( \lim_{{x \to 0}} x \sin\left(\frac{1}{x}\right) \)

To find: $$ \lim_{{x \to 0}} x \sin\left(\frac{1}{x}\right) $$ Step 1: Identify \( g(x) = x \sin\left(\frac{1}{x}\right) \). Step 2: Find bounding functions. Since \( -1 \leq \sin\left(\frac{1}{x}\right) \leq 1 \), multiply all parts by \( |x| \) to maintain inequalities: $$ -|x| \leq x \sin\left(\frac{1}{x}\right) \leq |x| $$ Here, \( f(x) = -|x| \) and \( h(x) = |x| \). Step 3: Evaluate the limits of the bounding functions: $$ \lim_{{x \to 0}} -|x| = 0 \quad \text{and} \quad \lim_{{x \to 0}} |x| = 0 $$ Step 4: By the Squeeze Theorem, $$ \lim_{{x \to 0}} x \sin\left(\frac{1}{x}\right) = 0 $$

Example 3: Evaluating \( \lim_{{x \to 0}} \sin(x)^{x} \)

To find: $$ \lim_{{x \to 0}} \sin(x)^{x} $$ This limit requires transformation: $$ \sin(x)^{x} = e^{x \ln(\sin(x))} $$ So, we need to evaluate: $$ \lim_{{x \to 0}} x \ln(\sin(x)) $$ Considering \( \sin(x) \approx x \) for small \( x \), we have: $$ \ln(\sin(x)) \approx \ln(x) $$ Thus: $$ \lim_{{x \to 0}} x \ln(x) = 0 $$ Therefore: $$ \lim_{{x \to 0}} \sin(x)^{x} = e^{0} = 1 $$

Theoretical Underpinnings of the Squeeze Theorem

The Squeeze Theorem relies on the concept of function bounding and continuity. It leverages the fact that if a function \( g(x) \) is confined between two functions \( f(x) \) and \( h(x) \) that converge to the same limit at a point \( c \), then \( g(x) \) must also converge to that limit at \( c \). This theorem is particularly valuable in cases where \( g(x) \) exhibits oscillatory behavior or lacks a straightforward limit evaluation strategy.

Common Applications of the Squeeze Theorem

The Squeeze Theorem finds application in various scenarios, including:

• Trigonometric Limits: Evaluating limits involving trigonometric functions that oscillate, such as \( \sin\left(\frac{1}{x}\right) \).
• Absolute Value Functions: Determining limits of functions bounded by absolute values, ensuring they remain within known limits.
• Indeterminate Forms: Resolving limits that initially present indeterminate forms like \( 0 \times \infty \) by constraining the function within manageable bounds.
• Oscillatory Functions: Handling functions that do not settle towards a single value without bounds, by narrowing their potential range.

Advantages and Limitations of the Squeeze Theorem

The Squeeze Theorem offers several advantages in limit evaluation:

• Versatility: Applicable to a wide range of functions, especially those involving oscillations or absolute values.
• Simplification: Provides a straightforward method when direct limit evaluation is complex or impossible.
• Theoretical Robustness: Strong foundation in real analysis ensures reliability in derived limits.

However, there are also limitations:

• Requirement of Bounding Functions: Necessitates the existence of suitable \( f(x) \) and \( h(x) \) that bound \( g(x) \), which may not always be identifiable.
• Symmetric Limits: Primarily effective for limits approaching a specific point and may not extend easily to infinite limits without modification.
• Dependent on Known Limits: The success of the theorem hinges on the ability to compute the limits of the bounding functions.

Comparison Table

Summary and Key Takeaways