Problem 37 - Calculus Early Transcendentals

Question

Prove that $$\lim _{x \rightarrow 0} x^{4} \cos \frac{2}{x}=0.$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Function

We need to find the limit of the function \( f(x) = x^4 \cos \frac{2}{x} \) as \( x \) approaches 0. This function includes \( x^4 \), which tends to 0 as \( x \to 0 \), and the oscillating term \( \cos \frac{2}{x} \), which ranges between -1 and 1.

Step 2: Applying Squeeze Theorem

The Squeeze Theorem is useful because we have the product of a bounded function (\( \cos \frac{2}{x}\)) with an expression that tends towards 0 (\( x^4 \)). Knowing that \(-1 \leq \cos \frac{2}{x} \leq 1\), we can create inequalities \(-x^4 \leq x^4 \cos \frac{2}{x} \leq x^4\).

Step 3: Analyzing the Extremes

We must evaluate the limits of the functions \(-x^4\) and \(x^4\) as \(x\) approaches 0. We have: \[ \lim_{x \to 0} -x^4 = 0 \] and \[ \lim_{x \to 0} x^4 = 0 \]. Thus, these bounding functions both tend to 0.

Step 4: Concluding with the Squeeze Theorem

Since \( -x^4 \leq x^4 \cos \frac{2}{x} \leq x^4 \) and both bounding functions converge to 0, the Squeeze Theorem implies that: \( \lim_{x \to 0} x^4 \cos \frac{2}{x} = 0 \).

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a useful tool in calculus, particularly when dealing with limits. Imagine you have a function trapped between two others. If you know the limiting behavior of these two outer functions, you can figure out the limit of the tricky one in the middle. In our example, the function

• 
  
  
  is trapped between
  -x^4
  
  and
  
  x^4
  
  , both of which have limits approaching 0 as
  
  x
  approaches 0. By the Squeeze Theorem,
  
  
  
  
  
  
  
  
  
  the limit of the middle function must also be 0. This theorem comes in handy, especially with oscillating functions like cosine, which complicate direct evaluation.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic and oscillate between fixed values. For cosine, the range is from -1 to 1 regardless of its argument. In our problem, we have

• 
  
  
  
  
  
  
  
  \( \cos \frac{2}{x} \) . The complicating factor is this quotient, which causes the cosine curve to oscillate wildly as \( x \) approaches zero.
  However, the key understanding here is the boundedness of cosine. This property allows other calculus tools, like the Squeeze Theorem, to step in and analyze the limit of our original function.

Limits and Continuity

Limits are the foundation of calculus, defining the behavior of functions as they approach particular points. A function might touch a point, or just hover near it, without ever landing precisely on that spot. In our scenario, we look at

• 
  \( \lim_{x \to 0} x^4 \cos \frac{2}{x} \) . The cosine oscillates between -1 and 1, but the factor \( x^4 \) shrinks towards zero. This makes the entire limit also zero as \( x \to 0 \) .
  Continuity means a function's graph is unbroken, which isn’t directly applicable here due to the cosine’s oscillations. Yet, by focusing on the limit, we still evaluate what happens as \( x \) drops to very small numbers, showcasing the interconnectedness of limits and continuity in calculus.

Calculus Proof Techniques

When proving limits, certain techniques are more effective than others. In this example, we used the Squeeze Theorem because of its power to corrale a bounded oscillating function. Let's break down how this proof technique functioned:

• 
  Identify the oscillating component like \( \cos \frac{2}{x} \) and understand its bounds.
  Set up inequalities with simpler bounding functions, specifically ones whose limits you can easily compute, such as \( -x^4 \leq x^4 \cos \frac{2}{x} \leq x^4 \).
  Calculate these easier limits where the bounding functions experience the same limiting behavior, \( \lim_{x \to 0} -x^4 = 0 \) and \( \lim_{x \to 0} x^4 = 0 \).
  Apply the Squeeze Theorem to conclude \( \lim_{x \to 0} x^4 \cos \frac{2}{x} = 0 \). This proof showcases how structured logical steps and the tools of calculus come together to address the challenge of limits with precision.