Problem 82 - Thomas Calculus

Question

a. Graph \(h(x)=x^{2} \cos \left(1 / x^{3}\right)\) to estimate \(\lim _{x \rightarrow 0} h(x),\) zooming in on the origin as necessary. b. Confirm your estimate in part (a) with a proof.

Step-by-Step Solution

Answer

\( \lim_{x \rightarrow 0} h(x) = 0 \).

Step 1: Understand the function

We need to graph the function \( h(x) = x^2 \cos\left(\frac{1}{x^3}\right) \). This function is defined for \( x eq 0 \) because \( \cos\left(\frac{1}{x^3}\right) \) is undefined at \( x = 0 \). The factor \( x^2 \) suggests that the function could approach zero as \( x \) approaches zero.

Step 2: Graph the Function

By using a graphing calculator or software like Desmos, plot the function \( h(x) = x^2 \cos\left(\frac{1}{x^3}\right) \). Zoom in on the graph around the origin (\( x = 0 \)) to observe the behavior of the function. You should see that as \( x \) approaches zero, the graph appears to approach \( h(x) = 0 \).

Step 3: Analyze the Behavior of the Function

Consider the function \( \cos\left(\frac{1}{x^3}\right) \), which oscillates between -1 and 1. The \( x^2 \) term, however, approaches zero as \( x \to 0 \). This suggests that regardless of the oscillating cosine term, the overall function \( h(x) \) should also approach 0 since \( x^2\cdot \text{bounded term} \to 0 \).

Step 4: Proof with Squeeze Theorem

Consider the inequality \(-x^2 \leq x^2 \cos\left(\frac{1}{x^3}\right) \leq x^2\). As \( x \rightarrow 0 \), both \(-x^2\) and \(x^2\) approach 0. By the squeeze theorem, since \( h(x) = x^2 \cos\left(\frac{1}{x^3}\right) \) is bound by these expressions, \( \lim_{x \to 0} h(x) = 0 \).

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a powerful tool in calculus used to determine the limit of a function. It's especially useful when dealing with functions that fluctuate or oscillate between two boundaries. Imagine you're trying to "squeeze" the function you're examining between two simpler functions that are easier to handle.

For the function \( h(x) = x^2 \cos\left(\frac{1}{x^3}\right) \), we can use the Squeeze Theorem by establishing the inequality

• \(-x^2 \leq x^2 \cos\left(\frac{1}{x^3}\right) \leq x^2\)

This is valid because the cosine function, \( \cos\left(\frac{1}{x^3}\right) \), always oscillates between -1 and 1. As \( x \) gets closer to 0, both \(-x^2\) and \(x^2\) approach 0 as well.
The theorem then tells us that if \(-x^2\) and \(x^2\) both tend toward the same limit, then \( h(x) \) must also tend to that limit. Thus,

• \( \lim_{x \to 0} h(x) = 0 \)

This method is precise and eliminates the guesswork, giving us a concrete proof for the behavior of the function \( h(x) \) at the limit point.

Function Behavior

Understanding the behavior of a function as it approaches a particular point is essential. This helps to intuitively grasp how the function behaves under different conditions.

For the function \( h(x) = x^2 \cos\left(\frac{1}{x^3}\right) \), consider the two components: \( x^2 \) and \( \cos\left(\frac{1}{x^3}\right) \).

• \( x^2 \) is straightforward; it becomes smaller and closer to zero as \( x \) approaches zero, regardless of any surrounding oscillations.
• The cosine part, \( \cos\left(\frac{1}{x^3}\right) \), fluctuates between -1 and 1. This rapid oscillation doesn't allow the cosine value itself to settle easily at a particular number.

Together, they form a function that, despite the wild oscillations, still becomes negligible because of the multiplicative factor \( x^2 \). As a result, the whole function \( h(x) \) approaches zero the closer \( x \) gets to zero.

By analyzing each component of the function, you gain insights into why the limit behaves the way it does, reinforcing the conclusion reached through formal methods like the Squeeze Theorem.

Graphing Functions

Graphing is an invaluable tool for visualizing function behavior and understanding how they approach certain limits.

To examine \( h(x) = x^2 \cos\left(\frac{1}{x^3}\right) \), graphing the function allows us to visually confirm its behavior near \( x = 0 \). By using graphing software like Desmos, you can see how \( h(x) \) behaves by zooming into the graph close to the origin.

• Initially, the graph might seem chaotic due to the oscillations of \( \cos\left(\frac{1}{x^3}\right) \), but as you zoom closer, it becomes evident that the function values tend toward zero.
• The graph effectively demonstrates what you've established analytically — that despite fluctuations, \( h(x) \) maintains a trajectory narrowing towards zero.

Using graphing techniques can help verify calculations and provide intuitive columns for complex mathematical concepts. It bridges the gap between abstract computation and tangible understanding, reinforcing the solution obtained through analytical means.