Problem 39 - Calculus: Early Transcendentals

Question

Prove that \( \displaystyle \lim_{x \to 0}x^4 \cos \frac{2}{x} = 0 \).

Step-by-Step Solution

Answer

The limit is 0 by the Squeeze Theorem.

Step 1: Understand the Problem Statement

We need to prove the limit \( \lim_{x \to 0} x^4 \cos\left(\frac{2}{x}\right) = 0 \). This means as \( x \) approaches 0, the expression \( x^4 \cos\left(\frac{2}{x}\right) \) should infinitely get closer to 0.

Step 2: Analyze the Trigonometric Component

The function \( \cos\left(\frac{2}{x}\right) \) oscillates between -1 and 1 for any real number \( x \). Therefore, \( -1 \leq \cos\left(\frac{2}{x}\right) \leq 1 \) for all \( x eq 0 \).

Step 3: Apply Absolute Value Properties

Given the properties of cosine, the absolute value of the product \( x^4 \cos\left(\frac{2}{x}\right) \) is bounded by the absolute value of \( x^4 \). Therefore, \(-x^4 \leq x^4 \cos\left(\frac{2}{x}\right) \leq x^4 \).

Step 4: Evaluate the Limit of Boundary Functions

Consider \( \lim_{x \to 0} -x^4 \) and \( \lim_{x \to 0} x^4 \). Since both \( -x^4 \) and \( x^4 \) approach 0 as \( x \to 0 \), by the Squeeze Theorem, we conclude that the limit of any function sandwiched between them will also be 0.

Step 5: Apply the Squeeze Theorem

Since \(-x^4 \leq x^4 \cos\left(\frac{2}{x}\right) \leq x^4 \) and both bounding functions tend to 0 as \( x \to 0 \), it follows by the Squeeze Theorem that \( \lim_{x \to 0} x^4 \cos\left(\frac{2}{x}\right) = 0 \).

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a powerful method for proving limits in calculus. It is particularly useful when dealing with functions that oscillate, like the cosine function. The theorem states that if a function is "squeezed" between two other functions that share the same limit at a certain point, the squeezed function must also approach that limit.
Let's break it down into simple terms:

• You have three functions: \( f(x) \), \( g(x) \), and \( h(x) \).
• For all values near a point \( a \), \( f(x) \leq g(x) \leq h(x) \).
• Both \( f(x) \) and \( h(x) \) have the same limit, \( L \), as \( x \) approaches \( a \).
• Then, \( g(x) \) must also have the limit \( L \) as \( x \to a \).

In our exercise, \( g(x) = x^4 \cos\left(\frac{2}{x}\right) \) is squeezed between \( f(x) = -x^4 \) and \( h(x) = x^4 \), both of which have a limit of 0 as \( x \to 0 \). Thus, by the Squeeze Theorem, \( \lim_{x \to 0} x^4 \cos\left(\frac{2}{x}\right) = 0 \).

Trigonometric Limits

Trigonometric limits often require special techniques like the Squeeze Theorem due to their oscillatory nature. Functions such as sine and cosine do not settle on a single value but instead fluctuate indefinitely between -1 and 1.
This behavior can complicate determining limits directly because, unlike polynomial or exponential functions, trigonometric functions do not converge to a particular point.
The key to evaluating these limits lies in understanding the properties of the functions:

• For \( \cos\left(\frac{2}{x}\right) \), the oscillation between -1 and 1 assures that it does not approach a single number as \( x \) approaches 0.
• Despite this, by multiplying \( \cos\left(\frac{2}{x}\right) \) with \( x^4 \), which does tend to 0, we can conclude using bounds that the overall expression approaches 0 as well.

Understanding these intricate behaviors allows students to employ suitable calculus techniques like the Squeeze Theorem more effectively.

Calculus Limit Theorems

Calculus limit theorems provide foundational tools for solving problems involving limits. Each theorem gives us insight into how functions behave as they approach certain values:

• Squeeze Theorem: Useful for functions caught between two simpler limits.
• Direct Substitution Theorem: Applies when inserting the value directly gives a definitive result.
• Limit Laws: Includes sum, product, and quotient laws for accommodating different function types.

These theorems enable us to maneuver through even the toughest limits by piecing together solutions from smaller, manageable parts. In our current exercise, we used the Squeeze Theorem to assess a trigonometric expression complicated by oscillation. By examining the boundaries and applying limits to simpler components, we harness the full range of these theorems to draw effective conclusions.
Each theorem leverages unique properties of functions, offering strategies that cater to their specific challenges. Mastery of these theorems is critical for any calculus student aiming to conquer a wide array of limit problems.