Discuss the continuity of cosine function [with Video] - Teachoo

Ex 5.1, 22 (i) - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.1, 22 (i) - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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

Transcript

Ex 5.1, 22 (i) Discuss the continuity of the cosine, cosecant, secant and cotangent functions.Let 𝒇(𝒙)=πœπ¨π¬β‘π’™ To check continuity of 𝑓(π‘₯), We check it’s if it is continuous at any point x = c Let c be any real number f is continuous at π‘₯ =𝑐 if if L.H.L = R.H.L = 𝑓(𝑐) i.e. lim┬(x→𝑐^βˆ’ ) 𝑓(π‘₯)= lim┬(x→𝑐^+ ) " " 𝑓(π‘₯)= 𝑓(𝑐) LHL at x β†’ c lim┬(x→𝑐^βˆ’ ) f(x) = lim┬(hβ†’0) f(c βˆ’ h) = lim┬(hβ†’0) cos⁑(π‘βˆ’β„Ž) = lim┬(hβ†’0) sin⁑𝑐 sinβ‘β„Ž+cos⁑𝑐 cosβ‘β„Ž Putting β„Ž=0 = sin⁑𝑐 sin⁑0+cos⁑𝑐 cos⁑0 = 0+cos c . 1 = 𝒄𝒐𝒔⁑𝒄 𝐴𝑠, cos⁑(π‘₯βˆ’π‘¦) =cos⁑π‘₯ cosβ‘π‘¦βˆ’sin⁑π‘₯ sin⁑𝑦 RHL at x β†’ c lim┬(x→𝑐^+ ) f(x) = lim┬(hβ†’0) f(c + h) = lim┬(hβ†’0) cos⁑(𝑐+β„Ž) = lim┬(hβ†’0) cos⁑𝑐 cosβ‘β„Ž – sin⁑𝑐 sinβ‘β„Ž Putting β„Ž=0 = cos⁑𝑐 cos⁑0 – sin⁑𝑐 sin⁑0 = cos c . 1 – 0 = 𝒄𝒐𝒔⁑𝒄 𝐴𝑠, cos⁑(π‘₯+𝑦) =cos⁑π‘₯ cosβ‘π‘¦βˆ’sin⁑π‘₯ sin⁑𝑦 And, 𝑓(𝑐) = cos⁑𝑐 Since, L.H.L = R.H.L = 𝑓(𝑐) ∴ Function is continuous at x = c Thus, we can write that f is continuous for x = c , where c βˆˆπ‘ ∴ 𝒄𝒐𝒔⁑𝒙 is continuous for every real number.

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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.