Problem 55 - 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. \(\lim _{x \rightarrow 1} \frac{\ln x^{2}}{\ln x}\)

Step-by-Step Solution

Answer

The limit exists and its value is 2.

Step 1: Understand the Expression

We need to find \( \lim_{{x \to 1}} \frac{\ln x^2}{\ln x} \). This involves the natural logarithm function, and the behavior as \( x \) approaches 1.

Step 2: Simplify the Expression

Observe that \( \frac{\ln x^2}{\ln x} \) can be rewritten using logarithm properties: \( \ln x^2 = 2\ln x \). Thus, the expression simplifies to \( \frac{2\ln x}{\ln x} = 2 \) for \( x eq 1 \).

Step 3: Analyze the Limit

Since the simplified form of the expression is a constant value 2 for any \( x eq 1 \), the limit as \( x \to 1 \) is not dependent on the value of \( \ln x \) at \( x = 1 \). Therefore, the limit exists, and its value is 2.

Step 4: Use a Table to Verify

To verify the limit, create a table with values of \( x \) approaching 1 from both sides (e.g., 0.9, 0.99, 0.999 and 1.1, 1.01, 1.001). Calculate \( \frac{\ln x^2}{\ln x} = 2 \) for these values. The table should consistently reflect a value approaching 2.

Step 5: Use Graphs for Further Verification

Construct a graph of the function \( y = \frac{\ln x^2}{\ln x} \). As \( x \) approaches 1 on the x-axis, the graph should remain at the constant value y = 2, confirming the table's numeric results.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It has some interesting properties that make it indispensable in calculus, particularly in dealing with limits and growth processes.
One crucial property is that the natural logarithm of a power can be simplified using the formula \( \ln \left( a^b \right) = b \ln a \).
This property was utilized in the solution to simplify \( \ln x^2 \) to \( 2 \ln x \).
This step is important as it makes the handling of the limit much simpler.

• Exponential and logarithmic functions are inverses.
• The derivative of \( \ln x \) is \( \frac{1}{x} \), which plays a role in calculus by helping in solving integration problems and evaluating limits.

Understanding these properties can significantly help in simplifying and solving limit problems involving natural logarithms.

Graphical Analysis

Graphical analysis provides a visual representation of functions and their limits. For the expression \( y = \frac{\ln x^2}{\ln x} \), graphing helps verify the limit especially when the algebraic simplification leads to a constant.
Here’s how graphical analysis can help in this case:
Plot the function on a graph. As \( x \) approaches 1 from either the left (values like 0.9, 0.99, 0.999) or the right (values like 1.1, 1.01, 1.001), the graph should confirm that \( y \) remains at 2.

• When algebraic manipulations provide a constant result, a graph will still visually affirm this behavior across different narrow intervals around the point of interest.
• Graphing removes uncertainty about algebraic simplification's accuracy and shows the behavior of the function in its domain.

This confirmation through graphs is an excellent way to solidify the understanding of a limit and check the work done manually or algebraically.

Limit Verification

Limit verification is a process used to ensure the accuracy of the determined limit. In this exercise, both a table and graph were used to verify the limit \( \lim_{{x \to 1}} \frac{\ln x^2}{\ln x} \).
Creating a table with values like 0.9, 0.99, 0.999 and 1.1, 1.01, 1.001, and evaluating \( \frac{\ln x^2}{\ln x} \) at these points helps visualize a trend approaching the constant value 2.
Here's how to verify limits effectively:

• Tables provide numerical results that show the limit's tendency as \( x \) nears the value.
• Graphs offer a visual cross-check for understanding how the function behaves across its domain.
• These methods together confirm that the limit exists and is, in this case, not contingent upon \( x = 1 \).

Verifying limits is crucial in calculus because it ensures rigor and confirms that mathematical simplifications (like reaching a constant) are correct.