Algebra of continous functions
Last updated at May 6, 2021 by Teachoo
Ex 5.1, 22 (i) Discuss the continuity of the cosine, cosecant, secant and cotangent functions.Let 𝒇(𝒙)=𝐜𝐨𝐬𝒙 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) sin𝑐 sinℎ+cos𝑐 cosℎ Putting ℎ=0 = sin𝑐 sin0+cos𝑐 cos0 = 0+cos c . 1 = 𝒄𝒐𝒔𝒄 𝐴𝑠, cos(𝑥−𝑦) =cos𝑥 cos𝑦−sin𝑥 sin𝑦 RHL at x → c lim┬(x→𝑐^+ ) f(x) = lim┬(h→0) f(c + h) = lim┬(h→0) cos(𝑐+ℎ) = lim┬(h→0) cos𝑐 cosℎ – sin𝑐 sinℎ Putting ℎ=0 = cos𝑐 cos0 – sin𝑐 sin0 = cos c . 1 – 0 = 𝒄𝒐𝒔𝒄 𝐴𝑠, cos(𝑥+𝑦) =cos𝑥 cos𝑦−sin𝑥 sin𝑦 And, 𝑓(𝑐) = cos𝑐 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 ∴ 𝒄𝒐𝒔𝒙 is continuous for every real number.