1. use the graph to determine the following limits.
(a) $\lim_{x \to 1} f(x)$
(b) $\lim_{x \to 2} f(x)$
(c) $\lim_{x \to 3} f(x)$
(d) $\lim_{x \to 4} f(x)$

2. use the graph to determine the following limits.
(a) $\lim_{x \to 1} f(x)$
(b) $\lim_{x \to 2} f(x)$
(c) $\lim_{x \to 3} f(x)$
(d) $\lim_{x \to 4} f(x)$

Question

1. use the graph to determine the following limits.
(a) $\lim_{x \to 1} f(x)$
(b) $\lim_{x \to 2} f(x)$
(c) $\lim_{x \to 3} f(x)$
(d) $\lim_{x \to 4} f(x)$

2. use the graph to determine the following limits.
(a) $\lim_{x \to 1} f(x)$
(b) $\lim_{x \to 2} f(x)$
(c) $\lim_{x \to 3} f(x)$
(d) $\lim_{x \to 4} f(x)$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Graphical Limits" ], "new_concepts": [], "current_concepts": [ "Graphical Limits", "One-Sided Limits" ] } ## Analyze limits for Question 1 Using the *Graphical Limits* knowledge point $$ \begin{aligned} &\text{At } x = 1: \lim_{x\to1^-} f(x) = 2, \lim_{x\to1^+} f(x) = 2 \implies \lim_{x\to1} f(x) = 2\ &\text{At } x = 2: \lim_{x\to2^-} f(x) = 1, \lim_{x\to2^+} f(x) = 1 \implies \lim_{x\to2} f(x) = 1\ &\text{At } x = 3: \lim_{x\to3^-} f(x) = -1, \lim_{x\to3^+} f(x) = 2 \implies \lim_{x\to3} f(x) \text{ does not exist (DNE)}\ &\text{At } x = 4: \lim_{x\to4^-} f(x) = 1, \lim_{x\to4^+} f(x) = 1 \implies \lim_{x\to4} f(x) = 1 \end{aligned} $$ ## Analyze limits for Question 2 Using the *Graphical Limits* knowledge point $$ \begin{aligned} &\text{At } x = 1: \lim_{x\to1^-} f(x) = 1, \lim_{x\to1^+} f(x) = 1 \implies \lim_{x\to1} f(x) = 1\ &\text{At } x = 2: \lim_{x\to2^-} f(x) = 1, \lim_{x\to2^+} f(x) = 2 \implies \lim_{x\to2} f(x) \text{ does not exist (DNE)}\ &\text{At } x = 3: \lim_{x\to3^-} f(x) = 2, \lim_{x\to3^+} f(x) = 2 \implies \lim_{x\to3} f(x) = 2\ &\text{At } x = 4: \lim_{x\to4^-} f(x) = 0, \lim_{x\to4^+} f(x) = 0 \implies \lim_{x\to4} f(x) = 0 \end{aligned} $$ ## Question 1 (a) $\lim_{x\to1} f(x) = 2$ (b) $\lim_{x\to2} f(x) = 1$ (c) $\lim_{x\to3} f(x)$ does not exist (DNE) (d) $\lim_{x\to4} f(x) = 1$ ## Question 2 (a) $\lim_{x\to1} f(x) = 1$ (b) $\lim_{x\to2} f(x)$ does not exist (DNE) (c) $\lim_{x\to3} f(x) = 2$ (d) $\lim_{x\to4} f(x) = 0$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Graphical Limits" ] }

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