suppose p and q are polynomial functions. if $\lim_{x \to 0} \frac{p(x)}{q(x)} = 8$ and $q(0) = 5$, find $p(0)$.

$p(0) = $ (simplify your answer. type an integer or a decimal.)

Question

suppose p and q are polynomial functions. if $\lim_{x \to 0} \frac{p(x)}{q(x)} = 8$ and $q(0) = 5$, find $p(0)$.

$p(0) = $ (simplify your answer. 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": [ "Polynomial Continuity", "Limit Laws", "Quotient Law for Limits" ], "new_concepts": [], "current_concepts": [ "Polynomial Continuity", "Limit Laws", "Quotient Law for Limits" ] } ## Apply limit laws to the quotient $$ \lim_{x \to 0} \frac{p(x)}{q(x)} = \frac{\lim_{x \to 0} p(x)}{\lim_{x \to 0} q(x)} = 8 $$ ## Use polynomial continuity to evaluate limits $$ \begin{aligned} &\lim_{x \to 0} p(x) = p(0) \ &\lim_{x \to 0} q(x) = q(0) = 5 \end{aligned} $$ ## Solve for the unknown value $$ \begin{aligned} &\frac{p(0)}{5} = 8 \ &p(0) = 40 \end{aligned} $$ Suppose $p$ and $q$ are polynomial functions. If $\lim_{x \to 0} \frac{p(x)}{q(x)} = 8$ and $q(0) = 5$, find $p(0)$. $p(0) =$ 40 (Simplify your answer. Type an integer or a decimal.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Limits of Rational Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Apply limit laws to the quotient

Use polynomial continuity to evaluate limits

Solve for the unknown value

Answer: