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Class 12 Continuity And Differentiability · Long Answer3 Markscontinuity_parameters

Class 12 Maths · Continuity And Differentiability · Long Answer #62

Let f(x)=[sin((a+1)x)+2sin x]/x for x<0, f(0)=2, and f(x)=[√(1+bx)−1]/x for x>0 near 0. Find a,b for continuity.

✅ Step-by-Step Solution

Final Answer: a=−1,b=4
  1. The left limit is (a+1)+2=a+3, using sin(cx)/x→c.
  2. Rationalize the right branch to b/[√(1+bx)+1], whose limit is b/2.
  3. For continuity both limits must equal 2, so a+3=2 and b/2=2.
  4. Therefore a=−1,b=4.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The left limit is (a+1)+2=a+3, using sin(cx)/x→c.
  • [1 mark] Rationalize the right branch to b/[√(1+bx)+1], whose limit is b/2.
  • [1 mark] For continuity both limits must equal 2, so a+3=2 and b/2=2.
  • [0.5 marks] Therefore a=−1,b=4.

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