use the graph to estimate to one decimal place the maximum value of $\delta$ corresponding to the given $\varepsilon$ satisfying the $\varepsilon$-$\delta$ definition of $\lim_{x \to c} f(x) = l$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Epsilon-Delta Definition" ], "new_concepts": [], "current_concepts": [ "Epsilon-Delta Definition", "Graphical Limit Estimation" ] } ## Identify the given values from the graph Using the **Epsilon-Delta Definition** knowledge point $$ \begin{aligned} &c = 3\ &L = 2\ &L + \varepsilon = 3 \implies \varepsilon = 1\ &L - \varepsilon = 1 \implies \varepsilon = 1 \end{aligned} $$ ## Determine the interval of x-values Using the **Epsilon-Delta Definition** knowledge point $$ \begin{aligned} &f(x) = L - \varepsilon = 1 \implies x_1 = 1\ &f(x) = L + \varepsilon = 3 \implies \text{No intersection (maximum of } f(x) \text{ is } 2.5 \text{ at } x = 5)\ &\text{Right boundary where } f(x) \ge 1 \implies x_2 = 9\ &x \in (1, 9) \implies L - \varepsilon < f(x) < L + \varepsilon \end{aligned} $$ ## Calculate the maximum delta Using the **Epsilon-Delta Definition** knowledge point $$ \begin{aligned} &\delta_1 = c - x_1 = 3 - 1 = 2\ &\delta_2 = x_2 - c = 9 - 3 = 6\ &\delta = \min(\delta_1, \delta_2) = \min(2, 6) = 2.0 \end{aligned} $$ Use the graph to estimate to one decimal place the maximum value of $\delta$ corresponding to the given $\varepsilon$ satisfying the $\varepsilon$-$\delta$ definition of $\lim_{x \to c} f(x) = L$. $\delta =$ 2.0 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Epsilon-Delta Definition" ] }

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Explanation:

Identify the given values from the graph

Determine the interval of x-values

Calculate the maximum delta

Answer: