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

 

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

 

Go Ad-free

Transcript

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

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.