in which limits below can we use lhospitals rule?

$\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}}$

$\lim_{x \to \infty} \frac{e^x}{x}$

$\lim_{x \to \pi/3} \frac{\sin x}{\pi/3 - x}$

$\lim_{x \to 0} \frac{\sin (3x)}{3x}$

Question

in which limits below can we use lhospitals rule?

$\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}}$

$\lim_{x \to \infty} \frac{e^x}{x}$

$\lim_{x \to \pi/3} \frac{\sin x}{\pi/3 - x}$

$\lim_{x \to 0} \frac{\sin (3x)}{3x}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Limits at Infinity" ], "new_concepts": [ "L'Hopital's Rule", "Indeterminate Forms" ], "current_concepts": [ "Limits at Infinity", "L'Hopital's Rule", "Indeterminate Forms" ] } ## Evaluate indeterminate forms for each limit $$ \begin{aligned} &\lim_{x\to\infty} \frac{\ln x}{\sqrt{x}} \implies \frac{\infty}{\infty} \quad (\text{Indeterminate form, L'Hospital's Rule applies})\ &\lim_{x\to\infty} \frac{e^x}{x} \implies \frac{\infty}{\infty} \quad (\text{Indeterminate form, L'Hospital's Rule applies})\ &\lim_{x\to\pi/3} \frac{\sin x}{\pi/3 - x} \implies \frac{\sin(\pi/3)}{0} = \frac{\sqrt{3}/2}{0} = \pm\infty \quad (\text{Not indeterminate, L'Hospital's Rule does not apply})\ &\lim_{x\to0} \frac{\sin(3x)}{3x} \implies \frac{0}{0} \quad (\text{Indeterminate form, L'Hospital's Rule applies}) \end{aligned} $$

(A) $\lim_{x\to\infty} \frac{\ln x}{\sqrt{x}}$

(B) $\lim_{x\to\infty} \frac{e^x}{x}$

(C) $\lim_{x\to\pi/3} \frac{\sin x}{\pi/3 - x}$

(D) $\lim_{x\to0} \frac{\sin(3x)}{3x}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "L'Hopital's Rule" ] }

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Question

Explanation:

Evaluate indeterminate forms for each limit

Answer: