Algebra of continous functions
Algebra of continous functions
Last updated at Aug. 13, 2021 by Teachoo
Ex 5.1, 20 Is the function defined by f (x) = ๐ฅ^2 โ sin x + 5 continuous at x = ฯ? f (x) = ๐ฅ^2 โ sin x + 5 Let ๐(๐ฅ)=๐ฅ^2 , ๐(๐ฅ)="sin x " & ๐(๐ฅ) = 5 ๐(๐) = ๐ฅ^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(๐ฅ)" ", ๐(๐ฅ) & ๐(๐ฅ) all are continuous at all real numbers then ๐(๐)= ๐(๐)โ๐(๐)+๐(๐) is continuous at "all real numbers" โด ๐(๐ฅ) = ๐ฅ^2 " โ sin x + 5" is continuous at all real numbers. Thus, ๐(๐ฅ) is continuous at ๐=๐ Ex 5.1, 20 (Method 1) Is the function defined by f (x) = ๐ฅ๏ทฎ2๏ทฏ โ sin x + 5 continuous at x = ฯ? f (x) = ๐ฅ๏ทฎ2๏ทฏ โ sin x + 5 We need to check continuity at ๐ฅ=๐ We know that A function f is continuous at ๐ฅ=๐ i.e. lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐ ๐๏ทฏ Thus lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐ ๐๏ทฏ โ f is continuous at ๐=๐ Ex 5.1, 20 (Method 2) Is the function defined by f (x) = ๐ฅ๏ทฎ2๏ทฏ โ sin x + 5 continuous at x = ฯ? f (x) = ๐ฅ๏ทฎ2๏ทฏ โ sin x + 5 Let ๐ ๐ฅ๏ทฏ= ๐ฅ๏ทฎ2๏ทฏ , ๐ ๐ฅ๏ทฏ=sin x & ๐ ๐ฅ๏ทฏ = 5 ๐ ๐ฅ๏ทฏ = ๐ฅ๏ทฎ2๏ทฏ is continuous as it is a polynomial ๐ ๐ฅ๏ทฏ=sin x is continuous at all real numbers ๐ ๐ฅ๏ทฏ = 5 is continuous as it is a constant function By Algebra of continuous function, If ๐ ๐ฅ๏ทฏ , ๐ ๐ฅ๏ทฏ & ๐ ๐ฅ๏ทฏ all are continuous at all real numbers then ๐ ๐ฅ๏ทฏ= ๐ ๐ฅ๏ทฏโ๐ ๐ฅ๏ทฏ+๐ ๐ฅ๏ทฏ is continuous at all real numbers โด ๐ ๐ฅ๏ทฏ = ๐ฅ๏ทฎ2๏ทฏ โ sin x + 5 is continuous at all real numbers. โ f is continuous at ๐=๐