evaluate the limit

\\lim_{x \to \infty} \frac{\sqrt{3 + 2x^2}}{(11 + 8x)}\

Question

evaluate the limit

\\lim_{x \to \infty} \frac{\sqrt{3 + 2x^2}}{(11 + 8x)}\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Limits at Infinity" ], "new_concepts": [], "current_concepts": [ "Limits at Infinity", "Rational Function Limits" ] } ## Identify the limit expression Using the *Limits at Infinity* knowledge point $$ L = \lim_{x \to \infty} \frac{\sqrt{3 + 2x^2}}{11 + 8x} $$ ## Divide numerator and denominator by x Using the *Limits at Infinity* knowledge point $$ \begin{aligned} L &= \lim_{x \to \infty} \frac{\frac{\sqrt{3 + 2x^2}}{x}}{\frac{11 + 8x}{x}} \ &= \lim_{x \to \infty} \frac{\sqrt{\frac{3 + 2x^2}{x^2}}}{\frac{11}{x} + 8} \ &= \lim_{x \to \infty} \frac{\sqrt{\frac{3}{x^2} + 2}}{\frac{11}{x} + 8} \end{aligned} $$ ## Evaluate the limit of each term Using the *Limits at Infinity* knowledge point $$ \begin{aligned} &\lim_{x \to \infty} \frac{3}{x^2} = 0 \ &\lim_{x \to \infty} \frac{11}{x} = 0 \ &L = \frac{\sqrt{0 + 2}}{0 + 8} = \frac{\sqrt{2}}{8} \end{aligned} $$ Evaluate the limit $$ \lim_{x \to \infty} \frac{\sqrt{3 + 2x^2}}{(11 + 8x)} = $$ $\frac{\sqrt{2}}{8}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Limits at Infinity" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the limit expression

Divide numerator and denominator by x

Evaluate the limit of each term

Answer: