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



  1. 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 sin⁑(π‘₯βˆ’π‘¦) =sin⁑π‘₯ cosβ‘π‘¦βˆ’cos⁑π‘₯ sin⁑𝑦 𝐴𝑠, cos⁑0=1 & sin⁑0=0 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‘₯+𝑦) =sin⁑π‘₯ cos⁑𝑦+cos⁑π‘₯ sin⁑𝑦 𝐴𝑠, cos⁑0=1 & sin⁑0=0 And, 𝑓(π‘₯) = sin⁑π‘₯ 𝑓(𝑐) = sin⁑𝑐 Hence, 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
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.