Problem 53 - 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.) $$f(x)=\frac{e^{2 x}-1}{x}$$

Step-by-Step Solution

Answer

The limit is 2; valid for \( x \in [-0.001, 0.001] \).

Step 1: Calculate Values of f(x)

To start, calculate the function values for the given inputs. For example, when \( x = 0.1 \), substitute into the function: \[ f(0.1) = \frac{e^{0.2} - 1}{0.1} \approx \frac{1.2214 - 1}{0.1} = 2.214. \]Repeat this calculation for all specified \(x\)-values: - \( f(0.01) = \frac{e^{0.02} - 1}{0.01} \approx 2.02 \)- \( f(0.001) \approx 2.002 \)- \( f(0.0001) \approx 2.0002 \)- \( f(-0.1) \approx 1.809 \)- \( f(-0.01) \approx 1.9801 \)- \( f(-0.001) \approx 1.998 \)- \( f(-0.0001) \approx 1.9998 \).

Step 2: Conjecture the Limit

Examine the pattern of the values calculated in Step 1. As \( x \) approaches zero from both positive and negative directions, \( f(x) \) seems to approach a particular value.The computed values suggest: - From the positive side: \( f(0.0001) \approx 2.0002 \)- From the negative side: \( f(-0.0001) \approx 1.9998 \)Thus, it appears that \( \lim_{x \to 0} f(x) = 2 \).

Step 3: Graph the Function

Use graphing software or a graphing calculator to plot the function \( f(x) = \frac{e^{2x} - 1}{x} \).Check the graph near \( x = 0 \) and observe if the value aligns with our limit conjecture.The graph should show \( f(x) \) approaches 2 closely as \( x \rightarrow 0 \).

Step 4: Determine an Interval for x

We need an interval where the difference between \( f(x) \) and the limit (2) is less than 0.01.For instance, find \( x \) where: \[ |f(x) - 2| < 0.01. \]From our calculations, - Checking \( x = 0.001 \): \( |2.002 - 2| = 0.002 < 0.01 \)- Checking \( x = -0.001 \): \( |1.998 - 2| = 0.002 < 0.01 \)Thus, an interval around \( x = 0 \) is approximately \([-0.001, 0.001]\).

Key Concepts

Function Values Table

Creating a function values table is a key step when examining limits in calculus. It involves calculating the output of a function for a range of input values, particularly those approaching the point of interest, like zero in this case. To illustrate this:

• Start by choosing values for your function input, such as small positive and negative fractions close to zero (e.g., 0.1, 0.01, 0.001, and so forth).
• Substitute each value into the function, computing the result. For example, for the function \( f(x) = \frac{e^{2x} - 1}{x} \), substituting \( x = 0.1 \) gives approximately \( f(0.1) = 2.214 \).
• Continue this process for all selected values to fill in the table.

This table helps us observe the behavior of the function as it approaches a critical point, providing insight for the next step: formulating a limit conjecture.

Limit Conjecture

A limit conjecture in calculus involves hypothesizing the expected value a function approaches as the input approaches a specific point. Based on the table from the previous step, observe how each calculated value behaves as it nears the target point, usually zero.
By examining our function values, we see that:

• For positive values nearing zero, \( f(x) \) inching closer to 2, such as \( f(0.0001) \approx 2.0002 \).
• For negative values nearing zero, \( f(x) \) also approaches 2, with \( f(-0.0001) \approx 1.9998 \).

Thus, one can conjecture that the limit of the function as \( x \) approaches zero from both sides is \( \lim_{x \to 0} f(x) = 2 \). This is an essential step in understanding the function's behavior and serves as a basis for further validation through graphing.

Graphing Functions

Graphing provides a visual representation of the function, offering a powerful tool to confirm our limit conjecture. The graph of a function like \( f(x) = \frac{e^{2x} - 1}{x} \) helps illustrate its behavior as \( x \) approaches zero.

• Using a graphing calculator or software, plot the function curve. This reveals how the function value trends near critical points.
• Pay close attention to the graph around \( x = 0 \). The graph should showcase the function converging towards the limit we speculated, in this case, 2.
• The overall slant or shape of the curve provides further insights into not only the approach to the limit but also how rapidly it converges.

Graphing solidifies the numerical observations made and can occasionally reveal discrepancies if the plotted curve does not align with our initial hypothesis.

Interval Calculation

Interval calculation near zero is crucial for verifying the conjectured limit with a specific tolerance. This involves identifying a range within which the function's output remains close to the conjectured limit.
Here's how to determine such an interval:

• Start by setting up an inequality to express the tolerance level. For example, finding when \( |f(x) - 2| < 0.01 \) means the function is within 0.01 of the limit value 2.
• By testing function values near zero, such as \( f(0.001) \approx 2.002 \) and \( f(-0.001) \approx 1.998 \), confirm they satisfy this condition since both are within 0.002 of 2.
• Declare this range as the interval: in this instance, \([-0.001, 0.001]\).

This interval ensures that for any \( x \) within the specified range, the function remains exceptionally close to the limit, confirming the stability of the limit conjecture.