Example 17 - Chapter 5 Class 12 Continuity and Differentiability

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Example 17 Discuss the continuity of sine function.Let 𝑓(π‘₯)=sin⁑π‘₯ Let’s check continuity of f(x) at any real number Let c be any real number. We know that A function is continuous at π‘₯ = 𝑐 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) = (π‘™π‘–π‘š)┬(β„Žβ†’0) sin⁑〖(π‘γ€—βˆ’β„Ž) = (π‘™π‘–π‘š)┬(β„Žβ†’0) (sin⁑𝑐 cosβ‘β„Ž "βˆ’ cos c sin h " ) = (sin⁑𝑐 cos⁑0 "βˆ’ cos c sin 0" ) = sin⁑𝑐× 1"βˆ’ cos c" Γ— 0 = sin c RHL at x β†’ c lim┬(x→𝑐^+ ) f(x) = lim┬(hβ†’0) f(c + h) = (π‘™π‘–π‘š)┬(β„Žβ†’0) sin⁑〖(𝑐〗+β„Ž) = (π‘™π‘–π‘š)┬(β„Žβ†’0) (sin⁑𝑐 cosβ‘β„Ž "+ cos c sin h " ) = (sin⁑𝑐 cos⁑0 "+ cos c sin 0" ) = sin⁑𝑐× 1" + cos c" Γ— 0 = sin c sin⁑(π‘₯βˆ’π‘¦) =sin⁑π‘₯ cosβ‘π‘¦βˆ’cos⁑π‘₯ sin⁑𝑦 sin⁑(π‘₯+𝑦) =sin⁑π‘₯ cos⁑𝑦+cos⁑π‘₯ sin⁑𝑦 𝐴𝑠, cos⁑0=1 & sin⁑0=0 𝐴𝑠, cos⁑0=1 & sin⁑0=0 And, 𝒇(𝒄) = π’”π’Šπ’β‘π’„ Since L.H.L = R.H.L = 𝑓(𝑐) Therefore, 𝑓(π‘₯) is continuous for all real number So, π’”π’Šπ’β‘π’™ is continuous.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo