Problem 47 - Calculus

Question

For the functions in Problems \(46-53,\) do the following: (a) Make a table of values of \(f(x)\) for \(x=0.1,0.01,0.001\) \(0.0001,-0.1,-0.01,-0.001,\) and -0.0001 (b) Make a conjecture about the value of \(\lim _{x \rightarrow 0} f(x)\) (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for \(x\) near 0 such that the difference between your conjectured limit and the value of the function is less than \(0.01 .\) (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.) $$f(x)=x^{2}-1$$

Step-by-Step Solution

Answer

(a) Table values trend to -1; (b) Limit is -1; (c) Graph confirms; (d) Interval \((-0.1, 0.1)\).

Step 1: Evaluate the Function at Given Points

To start, calculate the values of the function \( f(x) = x^2 - 1 \) for each specified \( x \) value.- For \( x = 0.1 \), \( f(0.1) = (0.1)^2 - 1 = 0.01 - 1 = -0.99 \)- For \( x = 0.01 \), \( f(0.01) = (0.01)^2 - 1 = 0.0001 - 1 = -0.9999 \)- For \( x = 0.001 \), \( f(0.001) = (0.001)^2 - 1 = 0.000001 - 1 = -0.999999 \)- For \( x = 0.0001 \), \( f(0.0001) = (0.0001)^2 - 1 = 0.00000001 - 1 = -0.99999999 \)- For \( x = -0.1 \), \( f(-0.1) = (-0.1)^2 - 1 = 0.01 - 1 = -0.99 \)- For \( x = -0.01 \), \( f(-0.01) = (-0.01)^2 - 1 = 0.0001 - 1 = -0.9999 \)- For \( x = -0.001 \), \( f(-0.001) = (-0.001)^2 - 1 = 0.000001 - 1 = -0.999999 \)- For \( x = -0.0001 \), \( f(-0.0001) = (-0.0001)^2 - 1 = 0.00000001 - 1 = -0.99999999 \)You can arrange these in a table for clarity.

Step 2: Make a Conjecture for the Limit

From the table of values, notice that as \( x \) approaches 0, \( f(x) \) approaches -1. Therefore, you can conjecture that:\[ \lim_{x \to 0} f(x) = -1 \]

Step 3: Graph the Function

Graph the function \( y = x^2 - 1 \). On the graph, notice how the curve forms a parabola opening upwards, with the vertex at \((0, -1)\). This visually confirms that as \( x \to 0 \), \( f(x) \to -1 \), which is consistent with our conjecture from Step 2.

Step 4: Find an Interval Around x=0 for Given Precision

We seek an interval around \( x = 0 \) such that \(|f(x) + 1| < 0.01\).- Start by setting up: \[ |x^2 - 1 + 1| < 0.01 \] - Simplify it: \[ |x^2| < 0.01 \]- This inequality results in: \[ -0.1 < x < 0.1 \] So an interval that satisfies this condition is \( -0.1 < x < 0.1 \). Here, the function values fall within the "window" of height 0.02 surrounding the limit value of -1 (from -1.01 to -0.99).

Key Concepts

Conjecturing Limits

When studying limits in calculus, particularly for functions approaching zero or any other number, we use the term "conjecture" to describe our best educated guess of a function's behavior as the input values get infinitely close to a point of interest. In this exercise, we are focusing on the function \( f(x) = x^2 - 1 \) as \( x \) approaches zero. By evaluating the function at various values very close to zero, such as \( x = 0.1, 0.01, 0.001 \), and their negative counterparts, we can see how \( f(x) \) behaves.
From the list provided, observe that the function values—\(-0.99, -0.9999, -0.999999\), etc.—draw nearer to \(-1\) as \( x \) decreases towards zero. This consistent pattern allows us to conjecture that the limit of \( f(x) \) as \( x \to 0 \) is within a whisper of \(-1\). Thus, we propose that the limit is exactly \(-1\), symbolically represented as \( \lim_{x \to 0} f(x) = -1 \). This conjecture becomes foundational to understanding limits.

Graphing Functions

Graphing functions is an essential part of validating our conjectures in calculus. It provides a visual proof of what we suspect to be true based on numerical or analytical methods. By graphing the function \( y = x^2 - 1 \), we gain insight into the behavior of the function around our point of interest, which is \( x = 0 \) in this case.
Interesting properties unfold when you draw its graph. The graph of \( y = x^2 - 1 \) forms an upward-opening parabola with its vertex at \( (0, -1) \). As \( x \) approaches zero from either direction, the output \( y \) nears \(-1\), further substantiating our earlier conjecture of the limit. Observing this parabola, we can also identify the symmetry about the \( y \)-axis, validating that changes in the sign of \( x \) do not impact the function's proximity to its limit at zero. Graphing assures that our conjectures are not only theoretically sound but visually convincing as well.

Evaluating Function Values

Actually evaluating function values for specific \( x \) can offer deep clarity in limits. This means computing \( f(x) \) for key points around the value you're interested in and analyzing the results. In the example with \( f(x) = x^2 - 1 \), calculating \( f(x) \) for progressively smaller (both positive and negative) values inches towards zero illustrates the function's behavior surrounding \( x = 0 \).
Consider how \( f(x) \) reacts when you specifically use inputs like \( x = 0.0001 \) or \( x = -0.0001 \). No matter the smallness of these numbers, the output \( f(x) \) returns results like \(-0.99999999\). These minuscule discrepancies indicate how nearly the function approaches its supposed limit of \(-1\), supporting our earlier conjectures. To encapsulate this concept: Evaluating function values decides the closeness of \( f(x) \) to a conjectured limit and whether this matches our graphical and theoretical insights.