a. evaluate $\lim_{x \to 1^+} \sqrt{x-1}$. b. explain why $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist. b. choose the correct answer below. a. $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is not defined for $x 1$. b. $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is not defined for $x 1$. d. $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is defined for $x

Question

a. evaluate $\lim_{x \to 1^+} \sqrt{x-1}$. b. explain why $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist. b. choose the correct answer below. a. $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is not defined for $x > 1$. b. $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is not defined for $x < 1$. c. $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is defined for $x > 1$. d. $\lim_{x \to 1^-} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is defined for $x < 1$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "One-Sided Limits", "Domain of Radical Functions" ], "new_concepts": [], "current_concepts": [ "One-Sided Limits", "Domain of Radical Functions" ] } ## Determine the domain of the radical function $$ \begin{aligned} x - 1 &\ge 0 \ x &\ge 1 \end{aligned} $$ ## Analyze the left-sided limit behavior $$ \begin{aligned} &\text{For } x \to 1^{-}, \text{ we consider values of } x < 1. \ &\text{Since } \sqrt{x-1} \text{ is undefined for } x < 1, \text{ the limit cannot be evaluated.} \end{aligned} $$ ## Match with the correct multiple-choice option $$ \lim_{x \to 1^{-}} \sqrt{x-1} \text{ does not exist because } \sqrt{x-1} \text{ is not defined for } x < 1. $$

(A) $\lim_{x \to 1^{-}} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is not defined for $x > 1$.

(B) $\lim_{x \to 1^{-}} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is not defined for $x < 1$.

(C) $\lim_{x \to 1^{-}} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is defined for $x > 1$.

(D) $\lim_{x \to 1^{-}} \sqrt{x-1}$ does not exist because $\sqrt{x-1}$ is defined for $x < 1$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "One-Sided Limits" ] }

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QUESTION IMAGE

Question

Explanation:

Determine the domain of the radical function

Analyze the left-sided limit behavior

Match with the correct multiple-choice option

Answer: