The graph provided is of the function $f(x) = \frac{1}{x}$. We are tasked with finding a number $\delta$ such that:

$|f(x) - 0.5| < 0.1$

whenever $0 < |x - 2| < \delta$.

Step-by-step process:

1. Express the condition: We need to satisfy the condition $|f(x) - 0.5| < 0.1$, where $f(x) = \frac{1}{x}$. Substituting this into the inequality, we get: $\left| \frac{1}{x} - 0.5 \right| < 0.1$

2. Simplify the inequality: To simplify this, we first rewrite the absolute value expression: $\left| \frac{1}{x} - 0.5 \right| = \left| \frac{1}{x} - \frac{1}{2} \right|$ This becomes: $\left| \frac{2 - x}{2x} \right| < 0.1$

3. Solve for $x$: To solve this inequality, we first multiply both sides by $2x$ (assuming $x > 0$, since the graph is valid for $x > 0$): $\left| 2 - x \right| < 0.2x$

4. Consider the two cases:

   • Case 1: $2 - x < 0.2x$ and
   • Case 2: $-(2 - x) < 0.2x$

Let's solve these two cases. I will calculate the solution. sympy import symbols, Eq, solve

Define variables

x = symbols('x')

Case 1: 2 - x < 0.2x

eq1 = Eq(2 - x, 0.2 * x) sol_case1 = solve(eq1, x)

Case 2: -(2 - x) < 0.2x

eq2 = Eq(-(2 - x), 0.2 * x) sol_case2 = solve(eq2, x)

sol_case1, sol_case2The solutions to the two cases are:

• For Case 1: $x = 1.67$
• For Case 2: $x = 2.5$

Interpretation of the results:

The inequality $|f(x) - 0.5| < 0.1$ holds when $x$ lies between these two values, i.e., for $1.67 < x < 2.5$.

Therefore, the value of $\delta$ should be the smallest distance from $x = 2$ to the boundary points 1.67 and 2.5. The closest boundary is at $x = 2.5$, and the distance from 2 to 2.5 is:

$\delta = 2.5 - 2 = 0.5$

Thus, $\delta = 0.5$.

Would you like further clarification or details on this solution?

Additional questions for you:

1. How do you verify the value of $\delta$ in real-world applications?
2. How would this problem change if the function were different, say $f(x) = x^2$?
3. What is the significance of the condition $|f(x) - 0.5| < 0.1$ in terms of the graph?
4. Can you extend this method to find a similar $\delta$ for any function $f(x)$?
5. How does the concept of limits relate to the condition $|f(x) - 0.5| < 0.1$?

Tip: When solving inequalities like this, it helps to break the absolute value into cases to simplify the expression!