1. Chapter 5 Class 12 Continuity and Differentiability
2. Serial order wise

Transcript

Ex 5.1, 25 Examine the continuity of f, where f is defined by ๐ ๐ฅ๏ทฏ= sin๏ทฎ๐ฅโ๐๐๐  ๐ฅ๏ทฏ, ๐๐ ๐ฅโ 0๏ทฎ&โ1, ๐๐ ๐ฅ=0๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ= sin๏ทฎ๐ฅโ๐๐๐  ๐ฅ๏ทฏ, ๐๐ ๐ฅโ 0๏ทฎ&โ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 โ cos x f(0) = sin 0 โ cos 0 = โ1 Hence, L H L = R H L = f(0) โด f (x) is continuous at x = 0 Case 2: For x โ  0 f(x) = sin x โ 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 ๐ฅ๏ทฏ both continuous for all real number , then f(x) = p(x) โ q(x) is continuous at all real numbers โด f(x) = sin x โ cos x is continuous for all real numbers. Hence, f(x) is continuous at all points