From the graph find the maximum δ>0 which is such that if 0<|x-4|<δ , then |x-2|<0.8
The graph is of f(x)=√x
To solve this problem, we need to find the maximum value of δ such that if 0 < |x - 4| < δ, then |x - 2| < 0.8.
From the given graph of f(x) = √x, we can visually see that as x approaches 4, the value of f(x) approaches 2. This means that we want to find the maximum value of δ such that if x is within a distance of δ from 4, the corresponding value of f(x) will be within a distance of 0.8 from 2.
To find this value numerically, we can use the properties of the square root function. Let's break down the inequalities step-by-step:
1. Start with the inequality |x - 4| < δ.
This means that x lies within a distance δ from 4. We can rewrite this as -δ < x - 4 < δ.
2. Substitute this x value into the inequality |x - 2| < 0.8.
We get -δ < (x - 2) < δ. Rearrange this to be -δ + 2 < x < δ + 2.
3. Combine the two inequalities to find the intersection:
Since -δ < x - 4 < δ and -δ + 2 < x < δ + 2, the possible range for x can be written as: max(-δ, -δ + 2) < x < min(δ, δ + 2).
4. Since we want the values of x to correspond to f(x) being within a distance of 0.8 from 2, we need to evaluate f(x) = √x within this possible range of x values.
5. We find f(max(-δ, -δ + 2)) and f(min(δ, δ + 2)), and set the difference between them to be less than or equal to 0.8.
In other words, we need to satisfy the inequality: |f(max(-δ, -δ + 2)) - f(min(δ, δ + 2))| ≤ 0.8.
By solving this inequality numerically, we can find the maximum value of δ that satisfies the given conditions.