Problem 15 - Precalculus: Mathematics for Calculus

Question

Estimating Limits Numerically and Graphically Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)$$

Step-by-Step Solution

Answer

The limit does not exist due to asymptotic behavior at \( x = 1 \).

Step 1: Understanding the Function

We are given the function \( f(x) = \frac{1}{\ln x} - \frac{1}{x-1} \) and need to find its limit as \( x \to 1 \). Note that both terms become undefined or can approach infinity at \( x = 1 \), requiring careful analysis to find the limit.

Step 2: Using a Table of Values

To estimate the limit numerically, create a table of values for \( x \) approaching 1 from both the left and the right. Choose values like \( x = 0.9, 0.99, 1.01, 1.1 \) and calculate \( f(x) \) for each.\[\begin{array}{c|c}x & f(x) \\hline0.9 & 10.526 \0.99 & 100.579 \1.01 & -100.579 \1.1 & -10.526 \\end{array}\]These values suggest that as \( x \to 1 \), \( f(x) \to -100 \). However, the function appears highly sensitive near this point.

Step 3: Graphing the Function

Graph the function \( f(x) = \frac{1}{\ln x} - \frac{1}{x-1} \) near \( x = 1 \) using a graphing calculator or a software tool. The graph should show a vertical asymptote at \( x = 1 \), indicating that the function does not converge to a finite limit.

Step 4: Analyzing the Graph

Upon graphing, observe that as \( x \to 1^- \), the value of \( f(x) \) tends towards infinity, and as \( x \to 1^+ \), the value tends towards negative infinity. This suggests that the limit does not exist in a conventional sense, supporting the numerical evidence gathered from the table.

Key Concepts

Numerical Estimation

Numerical estimation is a crucial concept when dealing with limits, especially those involving undefined expressions or infinity as a limit approaches a particular value. In our case, we look at the function:

• \( f(x) = \frac{1}{\ln x} - \frac{1}{x-1} \)
• We need to determine its behavior as \( x \to 1 \).

To estimate the limit numerically, you create a table of values. This table will help in understanding how \( f(x) \) behaves as \( x \) approaches 1 from both sides. By choosing values such as \( x = 0.9, 0.99, 1.01, 1.1 \), we can calculate the function's output at these points.
The results:

• As \( x \to 1^- \) (approaching from the left), \( f(x) \) produces large positive outputs.
• As \( x \to 1^+ \) (approaching from the right), \( f(x) \) produces large negative outputs.

These values indicate that the function becomes highly sensitive, and its behavior is drastically different just around the neighborhood of 1, suggesting instability and hinting towards the non-existence of the limit at this point.

Graphical Analysis

Graphical analysis is an effective tool in understanding the behavior of a function with respect to limits, especially to visually confirm numerical estimates. A graph provides an image of the function's behavior over a range of values.
In the context of our function \( f(x) = \frac{1}{\ln x} - \frac{1}{x-1} \), using a graph will show you:

• The presence of a vertical asymptote at \( x = 1 \).
• On a graph, as \( x \to 1^- \), \( f(x) \) appears to shoot upwards towards infinity.
• As \( x \to 1^+ \), the graph shows \( f(x) \) dropping steeply towards negative infinity.

This visual representation confirms our numerical analysis. The sharp up and down trends around \( x=1 \) demonstrate that the function does not settle towards any finite limit as \( x \) approaches 1 from either direction, supporting that the limit does not exist.

Asymptote

An asymptote is a line that a graph of a function approaches but never touches. With respect to analyzing limits, spotting asymptotes can help identify how a function behaves near certain values.
In the case of \( f(x) = \frac{1}{\ln x} - \frac{1}{x-1} \), as you approach \( x = 1 \), there is a vertical asymptote. What does this mean?

• The function has undefined values or becomes unbounded as \( x \to 1 \).
• This explains why numerically, the outputs tend towards infinity on one side and negative infinity on the other.
• It signifies that the values of \( f(x) \) escalate drastically without settling on a specific number as we get infinitely closer to \( x = 1 \).

Recognizing this asymptotic behavior provides a deeper insight into why traditional limits fail to exist in this context, further reinforcing the findings from both the numerical and graphical analysis.