We are asked to find a positive number $\delta$ such that for all $x$ satisfying $0 < |x - 1| < \delta$, it holds that $|f(x) - 1| < \frac{1}{2}$, where the function is $f(x) = \frac{1}{x}$.

Step-by-Step Solution:

We need to solve the inequality $|f(x) - 1| < \frac{1}{2}$, or equivalently, $\left| \frac{1}{x} - 1 \right| < \frac{1}{2}$.

1. Rewrite the absolute value inequality: $\left| \frac{1}{x} - 1 \right| = \left| \frac{1 - x}{x} \right| < \frac{1}{2}$ This inequality simplifies to: $\frac{|1 - x|}{|x|} < \frac{1}{2}$

2. Isolate $|1 - x|$: Multiply both sides of the inequality by $|x|$ (which is positive for $x > 0$): $|1 - x| < \frac{|x|}{2}$

3. Consider the neighborhood around $x = 1$: Since we are working around $x = 1$, we expect $x$ to be close to 1. Let’s assume $x$ is close enough to 1 so that $\frac{1}{2} \leq x \leq \frac{3}{2}$. This is a reasonable assumption because for values of $x$ far from 1, the inequality won't hold. So within this range: |x| \approx 1 \quad \text{(since x is close to 1)}

   Using this approximation, the inequality simplifies to: $|1 - x| < \frac{1}{2}$

4. Find $\delta$: The condition $|1 - x| < \frac{1}{2}$ corresponds directly to $\delta = \frac{1}{2}$, since $|x - 1| < \delta$ describes the width of the vertical strip around $x = 1$.

Thus, the value of $\delta$ is $\frac{1}{2}$.

Conclusion:

A positive number $\delta$ that satisfies the condition $| f(x) - 1 | < \frac{1}{2}$ for all $x$ such that $0 < |x - 1| < \delta$ is $\delta = \frac{1}{2}$.

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Do you have any questions or need further details?

Here are 5 related questions to consider:

1. How would the value of $\delta$ change if we required $|f(x) - 1| < \frac{1}{4}$?
2. What does the function $f(x) = \frac{1}{x}$ represent geometrically on a graph?
3. Can we find the $\delta$ for other values of $f(x)$, such as $f(x) = \frac{1}{x^2}$?
4. What would the behavior of $f(x)$ be for large $x$, and how does it affect $\delta$?
5. How does the choice of $\delta$ reflect the continuity of $f(x)$ near $x = 1$?

Tip: Whenever dealing with inequalities involving functions, rewriting the inequality in simpler terms can greatly help to visualize and solve the problem.