The value of k which makes the function defined by f (x) = {8(sin 1/x," if " x≠"0 " k      ", if x " ="0" )─ , continuous at x = 0 is

(A) 8 

(B) 1

(C) −1 

(D) None of these

This question is similar to Ex 5.1, 24 - Chapter 5 Class 12 - Continuity and Differentiability

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  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Serial order wise

Transcript

Question 7 The value of k which makes the function defined by f (x) = {β– 8(𝑠𝑖𝑛 1/π‘₯," if " π‘₯β‰ "0 " @π‘˜ ", if x " ="0" )─ , continuous at x = 0 is 8 (B) 1 (C) βˆ’1 (D) None of these At 𝒙 = 0 f(x) is continuous at π‘₯ =0 if L.H.L = R.H.L = 𝑓(0) if if lim┬(xβ†’0^βˆ’ ) 𝑓(π‘₯) = lim┬(xβ†’0^+ ) 𝑓(π‘₯) = 𝑓(0) LHL at x β†’ 0 lim┬(xβ†’0^βˆ’ ) f(x) = lim┬(hβ†’0) f(0 βˆ’ h) = lim┬(hβ†’0) f(βˆ’h) = lim┬(hβ†’0) sin⁑(1/(βˆ’β„Ž)) = (π’π’Šπ’Ž)┬(π‘β†’πŸŽ) γ€–βˆ’π’”π’Šπ’γ€—β‘(𝟏/𝒉) = (π‘™π‘–π‘š)┬(β„Žβ†’0) (βˆ’m) = βˆ’ m RHL at x β†’ 0 lim┬(xβ†’0^+ ) f(x) = lim┬(hβ†’0) f(0 + h) = lim┬(hβ†’0) f(h) = (π’π’Šπ’Ž)┬(π‘β†’πŸŽ) π’”π’Šπ’β‘(𝟏/𝒉) = (π‘™π‘–π‘š)┬(β„Žβ†’0) (m) = m ∴ L.H.L and R.H.L can never be equal as one is always negative of another. Hence, there does not exist any value of k for which f(x) is continuous at π‘₯=0 So, the correct answer is (D)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.