Examples

Chapter 5 Class 12 Continuity and Differentiability
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Example 17 Discuss the continuity of sine function.Let ๐(๐ฅ)=sinโก๐ฅ Letโs check continuity of f(x) at any real number Let c be any real number. We know that A function is continuous at ๐ฅ = ๐ if L.H.L = R.H.L = ๐(๐) i.e. limโฌ(xโ๐^โ ) ๐(๐ฅ)= limโฌ(xโ๐^+ ) " " ๐(๐ฅ)= ๐(๐) LHL at x โ c limโฌ(xโ๐^โ ) f(x) = limโฌ(hโ0) f(c โ h) = (๐๐๐)โฌ(โโ0) sinโกใ(๐ใโโ) = (๐๐๐)โฌ(โโ0) (sinโก๐ cosโกโ "โ cos c sin h " ) = (sinโก๐ cosโก0 "โ cos c sin 0" ) = sinโก๐ร 1"โ cos c" ร 0 = sin c RHL at x โ c limโฌ(xโ๐^+ ) f(x) = limโฌ(hโ0) f(c + h) = (๐๐๐)โฌ(โโ0) sinโกใ(๐ใ+โ) = (๐๐๐)โฌ(โโ0) (sinโก๐ cosโกโ "+ cos c sin h " ) = (sinโก๐ cosโก0 "+ cos c sin 0" ) = sinโก๐ร 1" + cos c" ร 0 = sin c sinโก(๐ฅโ๐ฆ) =sinโก๐ฅ cosโก๐ฆโcosโก๐ฅ sinโก๐ฆ sinโก(๐ฅ+๐ฆ) =sinโก๐ฅ cosโก๐ฆ+cosโก๐ฅ sinโก๐ฆ ๐ด๐ , cosโก0=1 & sinโก0=0 ๐ด๐ , cosโก0=1 & sinโก0=0 And, ๐(๐) = ๐๐๐โก๐ Since L.H.L = R.H.L = ๐(๐) Therefore, ๐(๐ฅ) is continuous for all real number So, ๐๐๐โก๐ is continuous.