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Magazine Chapter 3: Problem 115
Use input-output tables to find each limit. $$\lim _{x \rightarrow 1} \ln x$$
Short Answer
Expert verified
The limit is 0.
Step by step solution
01
- Understand the Limit
The problem asks to find the limit of \(\text{ln}(x)\) as \(x\) approaches 1. This means we need to see the behavior of the natural logarithm function near \(x = 1\), both from the left and the right.
02
- Create an Input-Output Table
Create a table with values of \(x\) close to 1 from both sides to see how \( \text{ln}(x) \) behaves. Choose values like 0.9, 0.95, 0.99, 1.01, 1.05, and 1.1.
03
- Compute Values
Calculate \( \text{ln}(x) \) for each chosen value of \(x\): \( x = 0.9 \rightarrow \text{ln}(0.9) 鈮� -0.105 x = 0.95 \rightarrow \text{ln}(0.95) 鈮� -0.051 x = 0.99 \rightarrow \text{ln}(0.99) 鈮� -0.010 x = 1.01 \rightarrow \text{ln}(1.01) 鈮� 0.010 x = 1.05 \rightarrow \text{ln}(1.05) 鈮� 0.049 x = 1.1 \rightarrow \text{ln}(1.1) 鈮� 0.095 \)
04
- Analyze the Results
From the table, observe that as \(x\) approaches 1 from both the left and the right, the values of \( \text{ln}(x) \) get closer to 0. For example, \( x = 0.99 \rightarrow \text{ln}(0.99) 鈮� -0.010\) and \(x = 1.01 \rightarrow \text{ln}(1.01) 鈮� 0.010\). This suggests the limit is 0.
05
- Conclusion
Based on the input-output table and analysis, conclude that \( \text{ln}(x) \) approaches 0 as \( x \) approaches 1. Therefore, \(\text{ln}(1) = 0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
natural logarithm
The natural logarithm, often denoted as \(\text{ln}(x)\), is a fundamental concept in calculus and mathematics. It is the logarithm to the base e, where e (\text{approximately 2.718}) is an important constant in mathematics due to its unique properties in growth and decay processes. The natural logarithm of a number y is the power to which e needs to be raised to obtain y. In other words, \[ \text{if } y = e^x, \text{ then } x = \text{ln}(y). \] Properties of the natural logarithm include:
- \(\text{ln}(1) = 0\)
- Inverse of the exponential function: \( e^{\text{ln}(x)} = x \)
- Logarithmic identity: \( \text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b) \)
input-output tables
Input-output tables are a helpful tool in analyzing the behavior of functions at specific points. They are particularly useful in finding limits, understanding function trends, and approximating values. An input-output table lists values of the independent variable (input) and the corresponding values of the dependent variable (output).
For instance, to find \( \text{lim}_{x \to 1} \text{ln}(x) \), we can create a table with inputs (x values) close to 1 to observe the behavior of \(\text{ln}(x)\). Steps might include:
For instance, to find \( \text{lim}_{x \to 1} \text{ln}(x) \), we can create a table with inputs (x values) close to 1 to observe the behavior of \(\text{ln}(x)\). Steps might include:
- Selecting values near 1, like 0.9, 0.95, 0.99, 1.01, 1.05, and 1.1
- Calculating the natural logarithm for each value
- Analyzing how outputs (natural logarithm values) change as inputs approach 1
behavior of functions at a point
Understanding the behavior of functions at a point is crucial in calculus. It helps in determining limits, continuity, and differentiability at that specific point. When examining the behavior of a function as it approaches a particular value, consider the following:
- \textbf{Approaching from Both Sides: } Evaluate the function using values slightly less and slightly more than the point in question (e.g., for 1, use 0.99 and 1.01).
- \textbf{Analyzing Trends: } Look for patterns or trends. For example, if values of \(\text{ln}(x)\) get closer to 0 as \(x \) approaches 1 from either side, it indicates that \( \text{lim}_{x \to 1} \text{ln}(x) = 0. \)
- \textbf{Graphical Representation: } Plotting the function can provide a visual understanding of how the function behaves around the point.聽
