Questions: The function (f) in the given figure satisfies (lim x rightarrow 3 f(x)=4). For each value of (varepsilon), find all values of (delta>0) such that (f(x)-4<varepsilon) whenever (0<x-3<delta). (a) (varepsilon=2) (b) (varepsilon=1) (c) For any (varepsilon), make a conjecture about the corresponding value of (delta) satisfying the above description.

The function (f) in the given figure satisfies (lim x rightarrow 3 f(x)=4). For each value of (varepsilon), find all values of (delta>0) such that (f(x)-4<varepsilon) whenever (0<x-3<delta).
(a) (varepsilon=2)
(b) (varepsilon=1)
(c) For any (varepsilon), make a conjecture about the corresponding value of (delta) satisfying the above description.

Transcript text: The function $f$ in the given figure satisfies $\lim _{x \rightarrow 3} f(x)=4$. For each value of $\varepsilon$, find all values of $\delta>0$ such that $|f(x)-4|<\varepsilon$ whenever $0<|x-3|<\delta$. (a) $\varepsilon=2$ (b) $\varepsilon=1$ (c) For any $\varepsilon$, make a conjecture about the corresponding value of $\delta$ satisfying the above description.

Solution

Solution Steps

Step 1: Understanding the Problem

We need to find the values of δ > 0 such that |f(x) - 4| < ε whenever 0 < |x - 3| < δ for given values of ε.

Step 2: Solving for ε = 2

For ε = 2, we need to find δ such that |f(x) - 4| < 2 whenever 0 < |x - 3| < δ.

From the graph, we observe that:

When x = 1, f(x) = 2, and |2 - 4| = 2.
When x = 5, f(x) = 6, and |6 - 4| = 2.

Thus, δ = 2 satisfies the condition for ε = 2.

Step 3: Solving for ε = 1

For ε = 1, we need to find δ such that |f(x) - 4| < 1 whenever 0 < |x - 3| < δ.

From the graph, we observe that:

When x = 2, f(x) = 3, and |3 - 4| = 1.
When x = 4, f(x) = 5, and |5 - 4| = 1.

Thus, δ = 1 satisfies the condition for ε = 1.

Final Answer

(a) For ε = 2, the inequality is 0 < δ ≤ 2.

(b) For ε = 1, the inequality is 0 < δ ≤ 1.

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