Answer:
To find a suitable delta, we need to analyze the graph of f(x) = √x. Let's start by plotting the graph of f(x) = √x:
```
^
| .
| .
| .
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|.
+------------------------------------------------------------>
0 3 6 9 12
```
The point we're interested in is (√x, x) = (3, 9), which corresponds to x = 9. We want to find a delta such that if |x - 9| < delta, then |√x - 3| < 0.4.
Let's consider the range of x-values that satisfy |x - 9| < delta. This translates to x being within a distance delta from 9 on the number line. Visually, this means considering the interval (9 - delta, 9 + delta) on the x-axis.
To ensure that |√x - 3| < 0.4, we need to find a delta such that the corresponding interval (9 - delta, 9 + delta) lies entirely within the interval (2.6, 3.4) on the y-axis.
From the graph, we can see that as x approaches 9, the corresponding y-values (√x) approach 3. So, we need to find a delta that guarantees that all x-values within (9 - delta, 9 + delta) will have corresponding y-values within (2.6, 3.4).
From the graph, we can estimate that the y-values will fall within the desired range if the x-values fall within (9 - delta, 9 + delta), where delta is approximately 0.4. Therefore, the appropriate delta would be 0.4.
Comparing the given options, we find that none of them match the estimated delta of 0.4. However, the closest option is:
B. 2.56
Please note that this is an estimate based on the graph, and a more precise calculation could be obtained through mathematical analysis.