Use the given graph to find 𝛿

Question

Use the given graph of f(x) = sqrt(x) to find the largest number  𝛿 such that

if 0 < | x-9 | < 𝛿 then | sqrt(x) - 3 | < 0.2

**the graph has points:

1. (?,2.8)
2. (9,3)
3. (?,3.2)

Sorry I cannot figure out how to attach the image of the graph.

Answer 1

From the graph it should be clear that |√x - 3| < 0.2 is true for exactly those x that

lie in the interval (2.8², 3.2²) = (7.84, 10.24) = (9 - 1.16, 9 + 1.24)

You can rephrase the question as finding the largest subinterval of the form

(9 - δ, 9 + δ).

To be a subinterval of (9 - 1.16, 9 + 1.24), δ cannot increase beyond 1.16.

Therefore that is the largest value of δ.