text: #### Please solve RD Sharma class 12 chapter Continuity exercise 8.2 question 3 subquestion (iii) maths textbook solution
  
  x = 0
  
  Hint:
  
  $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
  
  Given:
  
  $f(x)= \begin{cases}\frac{\sin x}{x} & x<0 \\ 2 x+3 & x \geq 0\end{cases}$
  
  Explanation:
  
  At x = 0
  
  \begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{\sin (-h)}{-h} \\ &=\lim _{h \rightarrow 0} \frac{-\sinh }{-h}=1 \end{aligned}
  
  \begin{aligned} &\text { R.H.L }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{\sinh }{h}=1 \\ &\text { Also } f(0)=2 \times 0+3=3 \\ &\text { Thus L.H.L}=\text {R.H.L} \neq \mathrm{f}(0) \end{aligned}
  
  f is discontinuous at x = 0
explanation: The document contains a math question from a textbook, specifically from RD Sharma class 12, chapter Continuity, exercise 8.2, question 3, subquestion (iii). The question involves evaluating the continuity of a piecewise function at \( x = 0 \). The document also provides a detailed solution to the problem.
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: #### Please solve RD Sharma class 12 chapter Continuity exercise 8.2 question 3 subquestion (iii) maths textbook solution
  
  x = 0
  
  Hint:
  
  $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
  
  Given:
  
  $f(x)= \begin{cases}\frac{\sin x}{x} & x<0 \\ 2 x+3 & x \geq 0\end{cases}$
  
  Explanation:
  
  At x = 0
  
  \begin{aligned} &\text { L.H.L }=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0}(0-h)=\lim _{h \rightarrow 0} \frac{\sin (-h)}{-h} \\ &=\lim _{h \rightarrow 0} \frac{-\sinh }{-h}=1 \end{aligned}
  
  \begin{aligned} &\text { R.H.L }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0}(0+h)=\lim _{h \rightarrow 0} \frac{\sinh }{h}=1 \\ &\text { Also } f(0)=2 \times 0+3=3 \\ &\text { Thus L.H.L}=\text {R.H.L} \neq \mathrm{f}(0) \end{aligned}
  
  f is discontinuous at x = 0
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question from a textbook, specifically from RD Sharma class 12, chapter Continuity, exercise 8.2, question 3, subquestion (iii). The question involves evaluating the continuity of a piecewise function at \( x = 0 \). The document also provides a detailed solution to the problem.
  
  Label: question-and-answer