evaluate the limit

\\lim_{x \to \infty} \frac{7x^3 - 6x^2 - 5x}{3 - 6x - 3x^3}\

Question

evaluate the limit

\\lim_{x \to \infty} \frac{7x^3 - 6x^2 - 5x}{3 - 6x - 3x^3}\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Limits at Infinity", "Rational Function Limits" ], "new_concepts": [], "current_concepts": [ "Limits at Infinity", "Rational Function Limits" ] } ## Identify the leading terms of the numerator and denominator $$ \begin{aligned} &\text{Numerator: } 7x^3 - 6x^2 - 5x \implies \text{Leading term: } 7x^3\ &\text{Denominator: } 3 - 6x - 3x^3 \implies \text{Leading term: } -3x^3 \end{aligned} $$ ## Divide the leading terms to evaluate the limit $$ \lim_{x \to \infty} \frac{7x^3 - 6x^2 - 5x}{3 - 6x - 3x^3} = \lim_{x \to \infty} \frac{7x^3}{-3x^3} = -\frac{7}{3} $$ Evaluate the limit $$\lim_{x \to \infty} \frac{7x^3 - 6x^2 - 5x}{3 - 6x - 3x^3}$$ $-\frac{7}{3}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Limits at Infinity" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the leading terms of the numerator and denominator

Divide the leading terms to evaluate the limit

Answer: