The problem asks us to use the graph of $f(x) = \sqrt{x}$ to find a value of $\delta$ such that $|\sqrt{x} - 2| < 0.4$ whenever $|x - 4| < \delta$.

Steps to solve:

1. Understanding the question:

   • The condition $|\sqrt{x} - 2| < 0.4$ means the value of $f(x) = \sqrt{x}$ is close to 2 (the function's value at $x = 4$, since $\sqrt{4} = 2$).
   • We need to find a range for $x$ around $x = 4$ such that the function value $f(x)$ remains within 0.4 units of 2.
   • Specifically, we want $\delta$ such that when $|x - 4| < \delta$, we get $|\sqrt{x} - 2| < 0.4$.

2. Interpreting the graph:

   • From the graph, we observe points where $f(x) = \sqrt{x}$. The graph indicates the behavior of the function for different values of $x$.
   • From the second image (or based on the graph interpretation), we see that $\sqrt{4} = 2$, and the goal is to find where $f(x)$ stays between 1.6 and 2.4 (i.e., $2 \pm 0.4$).

3. Finding the interval for $x$:

   • The points on the graph corresponding to $f(x) = 1.6$ and $f(x) = 2.4$ help us identify the range of $x$ values.
   • From the graph, it appears that $f(x) = 1.6$ when $x = 2$, and $f(x) = 2.4$ when $x = 6$.

4. Conclusion:

   • This means $f(x)$ lies between 1.