   1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
2. Serial order wise
3. Ex 5.1

Transcript

Ex 5.1, 22 (i) Discuss the continuity of the cosine, cosecant, secant and cotangent functions.Let 𝒇(𝒙)=𝐜𝐨𝐬⁡𝒙 To check continuity of 𝑓(𝑥), We check it’s if it is continuous at any point x = c Let c be any real number f is continuous at 𝑥 =𝑐 if if L.H.L = R.H.L = 𝑓(𝑐) i.e. lim┬(x→𝑐^− ) 𝑓(𝑥)= lim┬(x→𝑐^+ ) " " 𝑓(𝑥)= 𝑓(𝑐) LHL at x → c lim┬(x→𝑐^− ) f(x) = lim┬(h→0) f(c − h) = lim┬(h→0) cos⁡(𝑐−ℎ) = lim┬(h→0) sin⁡𝑐 sin⁡ℎ+cos⁡𝑐 cos⁡ℎ Putting ℎ=0 = sin⁡𝑐 sin⁡0+cos⁡𝑐 cos⁡0 = 0+cos c . 1 = 𝒄𝒐𝒔⁡𝒄 𝐴𝑠, cos⁡(𝑥−𝑦) =cos⁡𝑥 cos⁡𝑦−sin⁡𝑥 sin⁡𝑦 RHL at x → c lim┬(x→𝑐^+ ) f(x) = lim┬(h→0) f(c + h) = lim┬(h→0) cos⁡(𝑐+ℎ) = lim┬(h→0) cos⁡𝑐 cos⁡ℎ – sin⁡𝑐 sin⁡ℎ Putting ℎ=0 = cos⁡𝑐 cos⁡0 – sin⁡𝑐 sin⁡0 = cos c . 1 – 0 = 𝒄𝒐𝒔⁡𝒄 𝐴𝑠, cos⁡(𝑥+𝑦) =cos⁡𝑥 cos⁡𝑦−sin⁡𝑥 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 ∈𝐑 ∴ 𝒄𝒐𝒔⁡𝒙 is continuous for every real number. 