**Ex 5.1, 22**

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

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 provides courses for Mathematics from Class 9 to 12. You can ask questions here.