Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

Last updated at Dec. 8, 2016 by Teachoo

Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

Transcript

Ex 5.1, 22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions. (i) Let 𝒇 𝒙= 𝐜𝐨𝐬𝒙 Let c be any real number We know that A function is continuous at 𝑥 = 𝑐 if L.H.L = R.H.L = 𝑓 𝑐 i.e. limx→ 𝑐− 𝑓 𝑥= limx→ 𝑐+ 𝑓 𝑥= 𝑓 𝑐 Taking L.H.L limx→ 𝑐−𝑓 𝑥 = limx→ 𝑐− cos𝑥 Putting x = c − h = limx→ 𝑐− cos 𝑐−ℎ = limh→0 cos 𝑐−ℎ = limh→0 sin𝑐 sinℎ+ cos𝑐 cosℎ Putting ℎ=0 = sin𝑐 sin0+ cos𝑐 cos0 = 0+cos c . 1 = cos𝑐 Taking R.H.L limx→ 𝑐+𝑓 𝑥 = limx→ 𝑐+ cos𝑥 Putting x = c + h = limx→ 𝑐− cos 𝑐+ℎ = limh→0 cos 𝑐+ℎ = limh→0 cos𝑐 cosℎ – sin𝑐 sinℎ Putting ℎ=0 = cos𝑐 cos0 – sin𝑐 sin0 = cos c . 1 – 0 = cos𝑐 And, 𝑓 𝑥 = cos𝑥 𝑓 𝑐 = cos𝑐 Hence L.H.L = R.H.L = 𝑓 𝑐 ⇒ 𝑓 𝑥 is continuous for all real number So, 𝒄𝒐𝒔𝒙 is continuous. (ii) Let 𝒇 𝒙=cosec (x) 𝑓 𝑥 = 1 sin𝑥 Let 𝑝 𝑥=1 & 𝑞 𝑥= sin𝑥 Since, p(x) is a constant, ∴ 𝑝 𝑥 is continuous. We know that sin𝑥 is continuous for all real number ⇒ 𝑞 𝑥 is continuous . By Algebra of continuous function 𝑓 𝑥= 𝑝 𝑥𝑞 𝑥 , is continuous for all real numbers such that 𝑞 𝑥 ≠ 0 ⇒ 𝑓 𝑥 = 1 sin𝑥 is continuous for all real numbers such that sin𝑥 ≠ 0 i.e. 𝑥 ≠ 𝑛𝜋, 𝑛∈𝒁 So, 𝒄𝒐𝒔𝒆𝒄 𝒙 is continuous at all real numbers such that 𝒙 ≠ 𝒏𝝅, 𝒏∈𝒁 (iii) Let 𝒇 𝒙=sec x 𝑓 𝑥 = 1 c𝑜𝑠𝑥 Let 𝑝 𝑥=1 & 𝑞 𝑥= 𝑐𝑜𝑠𝑥 Since, p(x) is a constant, ∴ 𝑝 𝑥 is continuous. We know that cos𝑥 is continuous for all real number ⇒ 𝑞 𝑥 is continuous . By Algebra of continuous function 𝑓 𝑥= 𝑝 𝑥𝑞 𝑥 , is continuous for all real numbers such that 𝑞 𝑥 ≠ 0 ⇒ 𝑓 𝑥 = 1 cos𝑥 is continuous for all real numbers such that sin𝑥 ≠ 0 i.e. 𝑥 ≠ 2𝑛+1 𝜋2, 𝑛∈𝒁 So, 𝒔𝒆𝒄 𝒙 is continuous at all real numbers such that 𝒙 ≠ 𝟐𝒏+𝟏 𝝅𝟐, 𝒏∈𝒁 (iv) Let 𝒇 𝒙 = 𝒄𝒐𝒕 𝒙 𝑓 𝑥 = cos𝑥 sin𝑥 𝑓 𝑥 is defined for all real number except sin𝑥 = 0 i.e. x = 𝑛𝜋 Let 𝑝 𝑥= cos𝑥 & 𝑞 𝑥= sin𝑥 We know that sin𝑥 & cos x is continuous for all real number ⇒ p(x) & q (x) are continuous functions By Algebra of continuous functions, If 𝑝 𝑥 & 𝑞 𝑥 both continuous for all real number. then 𝑓 𝑥= 𝑝 𝑥𝑞 𝑥 continuous for all real numbers such that 𝑞 𝑥 ≠ 0 ⇒ 𝑓 𝑥 = cos𝑥 sin𝑥 is continuous for all real numbers such that sin𝑥 ≠ 0 i.e 𝑥 ≠ 𝑛𝜋, 𝑛∈𝒁 Hence, 𝐜𝐨𝒕𝒙 is continuous at all real numbers except 𝒙 ≠ 𝒏𝝅, 𝒏∈𝒁

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.