Problem 26 - Precalculus with Limits

Question

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 1} \dfrac{\textrm{ln}(x^2)}{x-1}$$

Step-by-Step Solution

Answer

This exercise's answer will depend on the specific values evaluated when constructing the table in Step 1. It's important to remember that the goal is to see what value the function is approaching as \(x\) gets closer to 1 from both sides, and to confirm this observation graphically.

Step 1: Create Table of Function Values

First, create a table of function values to estimate the limit numerically. Include x-values that approach 1 from both the left (like 0.9, 0.99, 0.999 etc.) and the right (like 1.1, 1.01, 1.001 etc.) and calculate the corresponding function values \(\frac{ln(x^2)}{x-1}\).

Step 2: Estimate the Limit

Look at the function values from the table generated in the previous step. As \(x\) approaches 1 from both sides, the function values should also be approaching a common number. This number is our estimated limit.

Step 3: Confirm the Result Graphically

Using a graphing utility, graph the function \(\frac{ln(x^2)}{x-1}\). Observing from the graph, as \(x\) gets closer to 1, the value of the function should also get closer to the limit estimated earlier. This confirms the limit graphically.

Key Concepts

Table of Values

Creating a table of values for a function is a useful step to explore how the function behaves near a particular point. In this exercise, you want to estimate the limit as values of x approach 1. This means you should choose x-values that are very close to 1, from both directions: the left and the right.

For example, select values like 0.9, 0.99, 0.999 for the left and 1.1, 1.01, 1.001 for the right. By calculating the expression \(\frac{\ln(x^2)}{x-1}\) for each of these x-values, you create a picture of the function's trend near the point x = 1.

• As x approaches 1 from the left, observe if the values increase or decrease.
• Similarly, do this for values approaching from the right.

If both sets of values are nearing the same number, this number is likely very close to the function's limit as x approaches 1. This tactic often gives a clearer numerical indication of what the limit might be, despite any algebraic complexity.

Graphical Confirmation

Once the table of values gives an estimated limit, it's important to verify your findings visually. This is where a graphing utility comes in handy. Graph the function \(\frac{\ln(x^2)}{x-1}\) to see its behavior as x approaches 1. When you graph this function, you'll likely observe how the y-values behave around x = 1.

• Look for a trend where the function seems to settle down to a particular y-value as x approaches 1 from both directions.
• Note if the curve stabilizes around your estimated limit. This agreement between the graph and your table of values is a great way to confirm your numerical estimate.

Using graphing not only confirms your result but also enhances your understanding of the function's behavior in context. This visual insight can sometimes reveal subtleties that numbers alone might miss.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.718281828. It's a fundamental concept in mathematics, especially in calculus and exponential growth and decay contexts.
In this exercise, \(\ln(x^2)\) plays a crucial role. recall some key properties of logarithms and how they apply to simplify expressions or solve equations.

• For example, \(\ln(x^2)\) can be rewritten using the power rule of logarithms: \(2 \ln(x)\).
• Understanding this conversion can simplify the original limit problem, providing a clearer path to finding its solution.

Comprehending how the natural logarithm functions and manipulates data is essential for tackling problems involving growth processes or continuous compounding. Thus, it's integral to grasp its behavior and properties thoroughly.