Problem 76 - Calculus

Question

For the functions in Problems \(76-77\), do the following: (a) Make a table of values of \(f(x)\) for \(x=a+0.1, a+0.01\) \(a+0.001, a+0.0001, a-0.1, a-0.01, a-0.001\) and \(a-0.0001\) (b) Make a conjecture about the value of \(\lim _{T \rightarrow a} f(x)\) (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for \(x\) containing \(a\) such that the difference between your conjectured limit and the value of the function is less than 0.01 on that interval. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.) $$f(x)=\frac{\cos 2 x-1+2 x^{2}}{x^{3}}, \quad a=0$$

Step-by-Step Solution

Answer

As \(x\to 0\), \(f(x)\) appears to approach \(-2\). The conjectured limit is \(-2\).

Step 1: Create a table of values

Calculate the value of \(f(x)\) for \(x = a + 0.1, a + 0.01, a + 0.001, a + 0.0001, a - 0.1, a - 0.01, a - 0.001, a - 0.0001\) where \(a = 0\). Substitute these values into the function \(f(x) = \frac{\cos(2x) - 1 + 2x^2}{x^3}\) and compute to get the approximations for limits.

Step 2: Analyze limit behavior

Observe the values in the table made in Step 1 to deduce the behavior of the function as \(x\) approaches 0. Look for a pattern that suggests the behavior of \(f(x)\) and allows you to conjecture the value of \(\lim _{x \rightarrow 0} f(x)\).

Step 3: Graph the function

Graph the function \(f(x) = \frac{\cos(2x) - 1 + 2x^2}{x^3}\) using a graphing tool. Check the graph to see if it supports the conjecture made in Step 2 about the limit.

Step 4: Find an interval for x

Determine an interval around \(a=0\) where the difference between \(f(x)\) and the conjectured limit is less than 0.01. This involves ensuring the graph exits the sides of a 0.02-height window, but not the top or bottom, suggesting \(|f(x) - L| < 0.01\), where \(L\) is your conjectured limit.

Key Concepts

Table of Values

A table of values is a crucial tool for understanding how a function behaves as its input values approach a specific point. For this exercise, the idea is to compute the function's values for several inputs that are very close to the point of interest, which is zero in this case. These inputs are selected symmetrically around zero, like \(0.1, 0.01, 0.001\), and also small deviations in the negative direction like \(-0.1, -0.01, -0.001\).
This helps us see any directional behavior of the function towards the point. By substituting these values into the function \(f(x) = \frac{\cos(2x) - 1 + 2x^2}{x^3}\), we can begin to observe how the function behaves as it gets closer to the target point of zero. This might include noticing whether the function heads towards a particular value (like 0, 1, -1, etc.) or whether it tends to infinity. Calculating these values can reveal underlying patterns, which are foundational for making predictions about limits.

Conjecture

A conjecture involves making an educated guess about the behavior of a function based on observed patterns. After you fill out the table of values for the function, observing these values can help you make a conjecture about the limit of the function as \(x\) approaches zero.
For example, if the values of \(f(x)\) seem to steer towards a particular number as \(x\) gets very close to 0, you might conjecture that the limit of \(f(x)\) as \(x\) approaches 0 is that number. It's important to note that a conjecture is not absolute truth, but rather a logical conclusion based on available evidence.
To form a solid conjecture, consider:\

• Consistency: Are the values consistently close to a certain number?
• Patterns: Do the values fluctuate wildly or more gently as they approach zero?

Remember, your conjecture sets the stage for further analysis, such as graphing and interval determination.

Graphing Functions

Graphing a function provides a visual representation that can validate or challenge your conjecture about the limit. In this exercise, graphing \(f(x) = \frac{\cos(2x) - 1 + 2x^2}{x^3}\) around \(x=0\) can help verify if the predictions were correct.
By plotting the function using a graphing calculator or graphing software, you can easily visualize:

• The behavior of the curve near the point of interest.
• Any asymptotic behavior as it approaches certain lines, suggesting it doesn't reach a finite height.
• Whether the curve supports the conjecture that the limit exists and corroborates what was observed in the table of values.

The graph might also show any potential discontinuities or unique behaviors at \(x=0\). Confirming your conjecture through this method provides a comprehensive understanding that is both numeric and graphical.

Interval Analysis

Interval analysis involves finding a range of x-values within which the function's output remains very close to the conjectured limit. In this exercise, you're trying to find an interval around zero where the function's values stay within 0.01 of the conjectured limit.
The process is as follows:

• Identify a range of x-values around zero, such as from \(x = -a\) to \(x = a\).
• Ensure that within this interval, the difference between \(f(x)\) and your conjectured limit is less than 0.01.

This essentially means that the graph of the function should exit through the vertical sides of a predefined window centered at the conjectured limit, without exiting through the top or bottom. This window's job is to highlight the region where the function behaves as expected, a key application of what we call epsilon-delta in formal limit proofs.
By establishing such an interval, you not only provide a more precise picture of how close the function gets to the limit but also underscore the predictability and reliability of your conjecture.