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Find the value(s) of k so that the following function is continuous at π‘₯ = 0
f (x) = { 1- cos ⁑kx / x sin⁑xΒ  if xβ‰ 0 1/2 if x=0

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Find the value(s) of k so that the following function is continuous

Question 21 - CBSE Class 12 Sample Paper for 2021 Boards - Part 2
Question 21 - CBSE Class 12 Sample Paper for 2021 Boards - Part 3
Question 21 - CBSE Class 12 Sample Paper for 2021 Boards - Part 4

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Question 21 Find the value(s) of k so that following function is continuous at π‘₯ = 0, f (x) = {β–ˆ((1 βˆ’ cosβ‘π‘˜π‘₯)/(π‘₯ sin⁑π‘₯ ) 𝑖𝑓 π‘₯β‰ 0@ 1/2 𝑖𝑓 π‘₯=0)─ Given that function is continuous at x = 0 𝑓(π‘₯) is continuous at x = 0 i.e. lim┬(xβ†’0) 𝑓(π‘₯)=𝑓(0) Limit at x β†’ 0 (π‘™π‘–π‘š)┬(π‘₯β†’0) f(x) = (π‘™π‘–π‘š)┬(β„Žβ†’0) f(h) = lim┬(hβ†’0) (1 βˆ’ cosβ‘π‘˜β„Ž)/(β„Ž (sinβ‘β„Ž) ) = lim┬(hβ†’0) (2 sin^2β‘γ€–π‘˜β„Ž/2γ€—)/(β„Ž (sinβ‘β„Ž)) = lim┬(hβ†’0) (2 sin^2β‘γ€–π‘˜β„Ž/2γ€—)/1 Γ—1/(β„Ž (sinβ‘β„Ž)) = lim┬(hβ†’0) (2 sin^2β‘γ€–π‘˜β„Ž/2γ€—)/(π‘˜β„Ž/2)^2 Γ— (π‘˜β„Ž/2)^2/(β„Ž (sinβ‘β„Ž)) = lim┬(hβ†’0) (2 sin^2β‘γ€–π‘˜β„Ž/2γ€—)/(π‘˜β„Ž/2)^2 Γ— (π‘˜^2 β„Ž^2)/(4β„Ž (sinβ‘β„Ž)) = lim┬(hβ†’0) (2 sin^2β‘γ€–π‘˜β„Ž/2γ€—)/(π‘˜β„Ž/2)^2 Γ— (π‘˜^2 β„Ž)/(4 (sinβ‘β„Ž)) = π‘˜^2/2 lim┬(hβ†’0) sin^2β‘γ€–π‘˜β„Ž/2γ€—/(π‘˜β„Ž/2)^2 Γ— β„Ž/sinβ‘β„Ž = π‘˜^2/2 lim┬(hβ†’0) sin^2β‘γ€–π‘˜β„Ž/2γ€—/(π‘˜β„Ž/2)^2 Γ—lim┬(hβ†’0) β„Ž/sinβ‘β„Ž = π‘˜^2/2 Γ— 1 Γ— 1 = π’Œ^𝟐/𝟐 Now, lim┬(xβ†’0) 𝑓(π‘₯)=𝑓(0) π‘˜^2/2 = 1/2 π‘˜^2 =1 π’Œ =±𝟏 Hence, k = 1, βˆ’1

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.