Ex 5.1, 20 - Ex 5.1

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

Transcript

Ex 5.1, 20 Is the function defined by f (x) = ๐‘ฅ^2 โ€“ sin x + 5 continuous at x = ฯ€? f (x) = ๐‘ฅ^2 โ€“ sin x + 5 Let ๐‘(๐‘ฅ)=๐‘ฅ^2 , ๐‘ž(๐‘ฅ)="sin x " & ๐‘Ÿ(๐‘ฅ) = 5 ๐’‘(๐’™) = ๐‘ฅ^2 is continuous as it is a polynomial ๐’’(๐’™)" = sin x" is continuous at all real numbers ๐’“(๐’™) = 5 is continuous as it is a constant function By Algebra of continuous functions, If ๐‘(๐‘ฅ)" ", ๐‘ž(๐‘ฅ) & ๐‘Ÿ(๐‘ฅ) all are continuous at all real numbers then ๐’‡(๐’™)= ๐’‘(๐’™)โˆ’๐’’(๐’™)+๐’“(๐’™) is continuous at "all real numbers" โˆด ๐‘“(๐‘ฅ) = ๐‘ฅ^2 " โ€“ sin x + 5" is continuous at all real numbers. Thus, ๐‘“(๐‘ฅ) is continuous at ๐’™=๐… Ex 5.1, 20 (Method 1) Is the function defined by f (x) = ๐‘ฅ๏ทฎ2๏ทฏ โ€“ sin x + 5 continuous at x = ฯ€? f (x) = ๐‘ฅ๏ทฎ2๏ทฏ โ€“ sin x + 5 We need to check continuity at ๐‘ฅ=๐œ‹ We know that A function f is continuous at ๐‘ฅ=๐œ‹ i.e. lim๏ทฎxโ†’๐œ‹๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ=๐‘“ ๐œ‹๏ทฏ Thus lim๏ทฎxโ†’๐œ‹๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ=๐‘“ ๐‘๏ทฏ โ‡’ f is continuous at ๐’™=๐… Ex 5.1, 20 (Method 2) Is the function defined by f (x) = ๐‘ฅ๏ทฎ2๏ทฏ โ€“ sin x + 5 continuous at x = ฯ€? f (x) = ๐‘ฅ๏ทฎ2๏ทฏ โ€“ sin x + 5 Let ๐‘ ๐‘ฅ๏ทฏ= ๐‘ฅ๏ทฎ2๏ทฏ , ๐‘ž ๐‘ฅ๏ทฏ=sin x & ๐‘Ÿ ๐‘ฅ๏ทฏ = 5 ๐‘ ๐‘ฅ๏ทฏ = ๐‘ฅ๏ทฎ2๏ทฏ is continuous as it is a polynomial ๐‘ž ๐‘ฅ๏ทฏ=sin x is continuous at all real numbers ๐‘Ÿ ๐‘ฅ๏ทฏ = 5 is continuous as it is a constant function By Algebra of continuous function, If ๐‘ ๐‘ฅ๏ทฏ , ๐‘ž ๐‘ฅ๏ทฏ & ๐‘Ÿ ๐‘ฅ๏ทฏ all are continuous at all real numbers then ๐‘“ ๐‘ฅ๏ทฏ= ๐‘ ๐‘ฅ๏ทฏโˆ’๐‘ž ๐‘ฅ๏ทฏ+๐‘Ÿ ๐‘ฅ๏ทฏ is continuous at all real numbers โˆด ๐‘“ ๐‘ฅ๏ทฏ = ๐‘ฅ๏ทฎ2๏ทฏ โ€“ sin x + 5 is continuous at all real numbers. โ‡’ f is continuous at ๐’™=๐…

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.