https://file.aiquickdraw.com/mathbot/file/1726277177127u6p3nqaq.jpg

The problem provided is about finding the largest \(\delta\) such that if \(0 < |x - 2| < \delta\), then \(|f(x) - 1| < 0.01\) for the given function \(f(x) = \frac{1}{x - 1}\). The graph of the function near \(x = 2\) is shown, and we need to use this to determine the value of \(\delta\).

### Step-by-Step Solution:

We are dealing with the function:
\[
f(x) = \frac{1}{x - 1}
\]
and we are asked to find the largest \(\delta\) such that:
\[
|f(x) - 1| < 0.01 \quad \text{whenever} \quad 0 < |x - 2| < \delta.
\]

1. **Rewriting the inequality:**
   The condition is \(|f(x) - 1| < 0.01\), which means:
   \[
   | \frac{1}{x - 1} - 1 | < 0.01.
   \]

   Simplifying the expression:
   \[
   \frac{1}{x - 1} - 1 = \frac{1 - (x - 1)}{x - 1} = \frac{2 - x}{x - 1}.
   \]
   So we now need:
   \[
   \left| \frac{2 - x}{x - 1} \right| < 0.01.
   \]

2. **Solving the inequality:**
   This inequality can be written as:
   \[
   \frac{|2 - x|}{|x - 1|} < 0.01.
   \]
   This implies that:
   \[
   |2 - x| < 0.01 |x - 1|.
   \]

3. **Graphical Interpretation:**
   We want to find the largest \(\delta\) such that when \(x\) is close to 2 (but not equal to 2), the function's values satisfy \(|f(x) - 1| < 0.01\). 

   The zoomed-in part of the graph near \(x = 2\) shows a range of values where \(f(x)\) is between 0.99 and 1.01. From this visual approximation, it seems that the interval around \(x = 2\) for which this condition holds is small. From the graph, we can estimate that \(\delta \approx 0.005\), since this is the range where the function is between 0.99 and 1.01.

Thus, the largest \(\delta\) is approximately **0.005** based on the graph.

---

Do you have any questions or need further details?

Here are 5 related questions to deepen understanding:
1. How can you calculate the exact value of \(\delta\) algebraically instead of graphically?
2. What does \(\lim_{x \to 2} f(x) = 1\) imply about the behavior of the function near \(x = 2\)?
3. How does the function behave as \(x\) approaches 1 from the right and left?
4. What are the implications of finding \(\delta\) in the context of limits and continuity?
5. How would the answer change if the inequality was \(|f(x) - 1| < 0.001\)?

**Tip:** Always double-check your results graphically and algebraically to ensure consistency.

The graph of the following function is shown in the figure. Find the largest δ such that if 0 < |x − 2| < δ, then |f(x) − 1| < 0.01 for f(x) = 1 / (x - 1).

Solve the problem of finding the largest δ such that 0 < |x − 2| < δ implies |f(x) − 1| < 0.01 for the function f(x) = 1 / (x - 1). A graphical and step-by-step algebraic explanation is provided for students studying calculus and limits.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation