Problem 52 - 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.) $$f(x)=\frac{e^{x}-1}{x}$$

Step-by-Step Solution

Answer

The limit \( \lim_{x \to 0} \frac{e^x - 1}{x} \) is 1.

Step 1: Evaluate Table of Values

To create a table of values for the function \(f(x) = \frac{e^{x} - 1}{x}\), substitute each given \(x\) value into the function and calculate \(f(x)\):\[\begin{array}{|c|c|}\hlinex & f(x) \\hline0.1 & \frac{e^{0.1} - 1}{0.1} \approx 1.0517 \0.01 & \frac{e^{0.01} - 1}{0.01} \approx 1.005 \0.001 & \frac{e^{0.001} - 1}{0.001} \approx 1.0005 \0.0001 & \frac{e^{0.0001} - 1}{0.0001} \approx 1.00005 \-0.1 & \frac{e^{-0.1} - 1}{-0.1} \approx 0.9516 \-0.01 & \frac{e^{-0.01} - 1}{-0.01} \approx 0.995 \-0.001 & \frac{e^{-0.001} - 1}{-0.001} \approx 0.9995 \-0.0001 & \frac{e^{-0.0001} - 1}{-0.0001} \approx 0.99995 \\hline\end{array}\]

Step 2: Conjecture the Limit

Based on the table of values, as \(x\) approaches 0 from both the positive and negative sides, the value of \(f(x)\) appears to approach 1. Thus, we can conjecture that:\[\lim_{x \to 0} \frac{e^{x} - 1}{x} = 1.\]

Step 3: Graph the Function

Graph \(f(x) = \frac{e^x - 1}{x}\) using graphing software or a graphing calculator. Observe the graph near \(x = 0\). The graph should show that as \(x\) approaches 0 from both sides, the function value approaches 1, supporting our conjecture from Step 2.

Step 4: Determine the Interval for Convergence

We want to find an interval \([-a, a]\) for \(x\) such that the absolute difference between the function \(f(x)\) and the conjectured limit 1 is less than 0.01, i.e., \[|f(x) - 1| < 0.01.\]Check values around \(x = 0\) until you find an interval such as \((-0.005, 0.005)\), where this condition holds. In this interval, the graph enters and exits through the sides of the window of height 0.02, not the top or bottom.

Key Concepts

Function Evaluation

When evaluating functions, especially those involving limits, we focus on determining the output of the function for specific input values of the variable. In this exercise, the function given is \( f(x) = \frac{e^x - 1}{x} \). Here, we substitute various small and large values of \( x \), both positive and negative, to observe the behavior of the function as \( x \) approaches zero.
We begin by substituting values close to zero such as 0.1, 0.01, 0.001, and even the negative counterparts. This yields a set of results that help in making observations about the nature of the function as the inputs become smaller. Calculating these values, as shown in the step-by-step solution, is essential as it forms the first piece in understanding the function's behavior near a specified point.

Graphing Functions

Graphing a function provides a visual representation, which is instrumental in corroborating our findings from numerical calculations. In the context of this exercise, graphing \( f(x) = \frac{e^x - 1}{x} \) around \( x = 0 \) allows us to see the function's behavior more clearly.

• The graph should visually confirm that as \( x \) nears zero, the line approaches a particular y-value without wavering wildly or sharply.
• With graphing software or a calculator, you can plot these points to generate the curve.
• Observing the graph will enable you to visualize the function approaching the horizontal line at \( y = 1 \), substantiating the numerical evidence obtained in the first step.

This graphical approach provides an intuitive grasp of the situation, confirming the predicted behavior through a different viewpoint.

Limit Conjecture

In mathematics, a limit conjecture involves predicting the behavior of a function as the input approaches a certain value. Here, with \( f(x) = \frac{e^x - 1}{x} \), the key is to determine \( \lim_{x \to 0} f(x) \).
From the function evaluations, both for positive and negative small \( x \)-values, it becomes evident that \( f(x) \) tends towards 1. Thus, we can conjecture that:

• \[ \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \]

This is a critical step in calculus, as it forms the backbone for more complex evaluations involving limits and continuity. The conjecture is supported both by algebraic calculations and visual confirmation via graphing.

Interval Estimation

Interval estimation in this context is about finding a range around \( x = 0 \) where the function's value remains very close to the conjectured limit. We specifically need the function \( f(x) \), within a small distance \( \delta \) about 0, to satisfy:
\( |f(x) - 1| < 0.01 \).
This determines how tight the function hugs the limit value 1 as \( x \) approaches zero. From numerical analysis or graph observations, you can estimate this window, which was found to be \((-0.005, 0.005)\). Within this interval, changes in \( x \) yield function values that stay comfortably within this bound, reflecting the stability of the conjectured limit.
Understanding interval estimation enhances our ability to deal with precision and approximation in real-world scenarios, ensuring calculated predictions remain reliable and applicable.