suppose $g(x) = f(7 - x)$ for all $x$, $\lim_{x \to 7^+} f(x) = 6$, and $\lim_{x \to 7^-} f(x) = 8$. find $\lim_{x \to 0^+} g(x)$ and $\lim_{x \to 0^-} g(x)$.

$\lim_{x \to 0^+} g(x) = $ (type an integer or a decimal.)

Question

suppose $g(x) = f(7 - x)$ for all $x$, $\lim_{x \to 7^+} f(x) = 6$, and $\lim_{x \to 7^-} f(x) = 8$. find $\lim_{x \to 0^+} g(x)$ and $\lim_{x \to 0^-} g(x)$.

$\lim_{x \to 0^+} g(x) = $ (type an integer or a decimal.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "One-Sided Limits" ], "new_concepts": [ "Limit Substitution" ], "current_concepts": [ "One-Sided Limits", "Limit Substitution" ] } ## Analyze the limit substitution $$ \text{Let } u = 7 - x. \text{ As } x \to 0^+, \text{ we have } x > 0. $$ $$ u = 7 - x \implies u < 7 \implies u \to 7^- $$ ## Evaluate the limit of g(x) $$ \lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} f(7 - x) = \lim_{u \to 7^-} f(u) $$ ## Substitute the given limit value $$ \lim_{u \to 7^-} f(u) = 8 $$ Suppose $g(x) = f(7 - x)$ for all $x$, $\lim_{x \to 7^+} f(x) = 6$, and $\lim_{x \to 7^-} f(x) = 8$. Find $\lim_{x \to 0^+} g(x)$ and $\lim_{x \to 0^-} g(x)$. $\lim_{x \to 0^+} g(x) =$ 8 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "One-Sided Limits" ] }

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Question

Explanation:

Analyze the limit substitution

Evaluate the limit of g(x)

Substitute the given limit value

Answer: