Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{e^{-x}-1}{x}\)

Step-by-Step Solution

Answer

The limit exists and its value is 1.

Step 1: Understanding the Problem

We need to find the limit \( \lim _{x \rightarrow 0} \frac{e^{-x}-1}{x} \). This involves checking the behavior of the function \( f(x) = \frac{e^{-x}-1}{x} \) as \( x \) approaches 0. We aim to determine whether this limit exists and, if it does, find its value.

Step 2: Creating a Table of Values

To explore the behavior of \( f(x) \) as \( x \) approaches 0, let's calculate the function values for \( x \) close to 0. Consider values such as \( x = -0.1, -0.01, 0.01, 0.1 \) and so on. Fill in a table with these \( x \) values and corresponding \( f(x) \) values.

Step 3: Calculating Function Values

By calculating, we find:- For \( x = -0.1 \), \( f(-0.1) \approx 1.0517 \)- For \( x = -0.01 \), \( f(-0.01) \approx 1.0050 \)- For \( x = 0.01 \), \( f(0.01) \approx 0.9950 \)- For \( x = 0.1 \), \( f(0.1) \approx 0.9516 \)The function values seem to approach 1 as \( x \to 0 \) from both sides.

Step 4: Plotting a Graph

Plotting \( f(x) = \frac{e^{-x}-1}{x} \) will allow us to visualize the behavior of the function near \( x = 0 \). In the graph, observe that as \( x \) approaches 0 from both left and right, the function values trend towards a particular value.

Step 5: Analyzing the Graph

Upon graph analysis, the function values confirm the trend seen in the table. The function \( f(x) \) approaches 1 as \( x \to 0 \) from both sides, which indicates that the limit exists.

Step 6: Conclusion

Based on the table and graph, we conclude that the limit exists and is 1. Thus, \( \lim _{x \rightarrow 0} \frac{e^{-x}-1}{x} = 1 \).

Key Concepts

Table of Values

Understanding limits often starts with a table of values. Here, we focus on our function \( f(x) = \frac{e^{-x} -1}{x} \). By choosing specific values for \( x \) close to the point of interest, 0 in this case, we can see how the function behaves as it approaches that point.

• Choose values like \( x = -0.1, -0.01, 0.01, 0.1 \).
• Calculate \( f(x) \) for each \( x \).

For example, by plugging \( x = -0.1 \), we get \( f(-0.1) \approx 1.0517 \), while with \( x = 0.01 \), \( f(0.01) \approx 0.9950 \). You'll notice a pattern where the values seem to get closer to 1.
It's like having snapshots of the function near a critical point, showing us that as \( x \to 0 \), our function value approaches 1.

Function Behavior

Next, let’s delve into how the function behaves as \( x \) moves towards zero. The expression \( \frac{e^{-x} - 1}{x} \) is inherently tricky to evaluate directly at \( x = 0 \) because it looks like a \( \frac{0}{0} \) form at first glance. This is where understanding behavior helps.
From the table of values, you might observe that approaching from either negative or positive directions, i.e., as \( x \) becomes very small in magnitude, leads our function to head towards a single value.

• For slightly negative \( x \), values like 1.0517 were noted.
• For slightly positive \( x \), values like 0.9950 were noted.

The more refined our x values get, the more exact our function's predictable pattern becomes. This pattern hitting around 1 indicates the existence of the limit as partitions from both sides agree at \( x \to 0 \). This converging trend is key to concluding that \( \lim _{x \to 0} \frac{e^{-x} - 1}{x} = 1 \).

Graph Analysis

While numbers from a table give a discrete look, graph analysis provides a continuous insight into function behavior around the limit's point. Plotting \( f(x) = \frac{e^{-x} - 1}{x} \) reveals a visual story.
Here’s how:

• As you draw the function, you'll see the curve approaching a horizontal line as it nears the y-axis.
• The line near \( y = 1 \) indicates where function values hover around \( x = 0 \).
• The slope of the curve steadies at the approach to this particular y-value.

Graphs can highlight any anomalies or unexpected dips but in this function’s case, you observe smooth transitions on both sides of \( x = 0 \).
This visual check bolsters the calculations and table observations, confirming our calculated and potentially abstract conclusions that \( \lim_{x \to 0} \frac{e^{-x} - 1}{x} \) is indeed 1.