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Ex 5.1, 25 - Examine continuity of f(x) = {sin x - cos x, -1 - Ex 5.1

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.1, 25 Examine the continuity of f, where f is defined by ๐‘“ ๐‘ฅ๏ทฏ= sin๏ทฎ๐‘ฅโˆ’๐‘๐‘œ๐‘  ๐‘ฅ๏ทฏ, ๐‘–๐‘“ ๐‘ฅโ‰ 0๏ทฎ&โˆ’1, ๐‘–๐‘“ ๐‘ฅ=0๏ทฏ๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ= sin๏ทฎ๐‘ฅโˆ’๐‘๐‘œ๐‘  ๐‘ฅ๏ทฏ, ๐‘–๐‘“ ๐‘ฅโ‰ 0๏ทฎ&โˆ’1, ๐‘–๐‘“ ๐‘ฅ=0๏ทฏ๏ทฏ Case 1: At x = 0 f (x) is continuous at x = 0 if LHL = RHL = f(0) l๐‘ก๏ทฎxโ†’0๏ทฏ๐‘“ ๐‘ฅ๏ทฏ = l๐‘ก๏ทฎxโ†’0๏ทฏ๐‘“ ๐‘ฅ๏ทฏ = ๐‘“ 0๏ทฏ And f(x) = sin x โˆ’ cos x f(0) = sin 0 โˆ’ cos 0 = โˆ’1 Hence, L H L = R H L = f(0) โˆด f (x) is continuous at x = 0 Case 2: For x โ‰  0 f(x) = sin x โˆ’ cos x Let p (x) = sinx & q (x) = cos x We know that, sin x & cos x are both continuous functions So, p(x) & q (x) is continuous at all real numbers By Algebra of continuous functions, If ๐‘ ๐‘ฅ๏ทฏ & ๐‘ž ๐‘ฅ๏ทฏ both continuous for all real number , then f(x) = p(x) โˆ’ q(x) is continuous at all real numbers โˆด f(x) = sin x โˆ’ cos x is continuous for all real numbers. Hence, f(x) is continuous at all points

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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