text: # $$\rm \displaystyle \lim_{x \rightarrow \infty} 2x \sin \left(\frac{4} {x}\right)$$ का मान ज्ञात कीजिए।
  
  1. 2
  2. 4
  3. 8
  4. $$\frac 1 2$$
  
  Option 3 : 8
  Free
  CRPF Head Constable Ministerial 22 Feb 2023 (Shift 1) Memory-Based Test
  1.3 Lakh Users
  100 Questions 100 Marks 90 Mins
  
  ## Detailed Solution
  
  संकल्पना:
  
  $$\rm \displaystyle \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$$
  
  गणना:
  
  $$\rm \displaystyle \lim_{x \rightarrow \infty} 2x \sin \left(\frac{4} {x}\right)$$
  
  $$\rm 2 \times \displaystyle \lim_{x → ∞} \frac{\sin \left(\frac{4} {x}\right)}{\left(\frac{1}{x} \right )}$$
  
  $$\rm 2 \times \displaystyle \lim_{x → ∞} \frac{\sin \left(\frac{4} {x}\right)}{\left(\frac{4}{x} \right )} \times 4$$
  
  माना कि $$\rm \frac {4}{x} = t$$ है।
  
  यदि x → ∞ है, तो t → 0 है।
  
  $$\rm 8 \times\displaystyle \lim_{t \rightarrow 0} \frac{\sin t}{t}$$
  
  = 8 × 1
  
  = 8
explanation: The document contains a math question and its detailed solution, including the correct answer. The question is about finding the limit of a function as \( x \) approaches infinity, and the correct answer is provided as 8.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # $$\rm \displaystyle \lim_{x \rightarrow \infty} 2x \sin \left(\frac{4} {x}\right)$$ का मान ज्ञात कीजिए।
  
  1. 2
  2. 4
  3. 8
  4. $$\frac 1 2$$
  
  Option 3 : 8
  Free
  CRPF Head Constable Ministerial 22 Feb 2023 (Shift 1) Memory-Based Test
  1.3 Lakh Users
  100 Questions 100 Marks 90 Mins
  
  ## Detailed Solution
  
  संकल्पना:
  
  $$\rm \displaystyle \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$$
  
  गणना:
  
  $$\rm \displaystyle \lim_{x \rightarrow \infty} 2x \sin \left(\frac{4} {x}\right)$$
  
  $$\rm 2 \times \displaystyle \lim_{x → ∞} \frac{\sin \left(\frac{4} {x}\right)}{\left(\frac{1}{x} \right )}$$
  
  $$\rm 2 \times \displaystyle \lim_{x → ∞} \frac{\sin \left(\frac{4} {x}\right)}{\left(\frac{4}{x} \right )} \times 4$$
  
  माना कि $$\rm \frac {4}{x} = t$$ है।
  
  यदि x → ∞ है, तो t → 0 है।
  
  $$\rm 8 \times\displaystyle \lim_{t \rightarrow 0} \frac{\sin t}{t}$$
  
  = 8 × 1
  
  = 8
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its detailed solution, including the correct answer. The question is about finding the limit of a function as \( x \) approaches infinity, and the correct answer is provided as 8.
  
  Label: question-and-answer