Ex 5.1, 1 - Prove f(x) = 5x - 3 is continuous at x = 0, -3 - Checking continuity at a given point

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.1, 1 Prove that the function ๐‘“ (๐‘ฅ) = 5๐‘ฅ โ€“ 3 is continuous at ๐‘ฅ = 0, at ๐‘ฅ = โ€“ 3 and at ๐‘ฅ = 5 Given ๐‘“ (๐‘ฅ) = 5๐‘ฅ โ€“ 3 (i) At ๐’™=๐ŸŽ f is continuous at x = 0 if ๏ท๐ฅ๐ข๐ฆ๏ทฎ๐ฑโ†’๐ŸŽ๏ทฏ ๐’‡(๐’™) = ๐’‡(๐ŸŽ) Since, L.H.S = R.H.S โˆด ๏ทlim๏ทฎxโ†’0๏ทฏ ๐‘“(๐‘ฅ) = ๐‘“(0) Hence, f is continuous at ๐’™ = ๐ŸŽ (ii) At x = โˆ’3 f is continuous at x = โˆ’3 if ๏ท lim๏ทฎxโ†’โˆ’3๏ทฏ ๐‘“๏ท๐‘ฅ๏ทฏ= ๐‘“๏ทโˆ’3๏ทฏ Since, L.H.S = R.H.S โˆด ๏ทlim๏ทฎxโ†’โˆ’3๏ทฏ ๐‘“(๐‘ฅ) = ๐‘“(โˆ’3) Hence, f is continuous at ๐’™ =โˆ’3 (iii) At ๐’™ =๐Ÿ“ f is continuous at x = 5 if ๏ท lim๏ทฎxโ†’5๏ทฏ ๐‘“๏ท๐‘ฅ๏ทฏ= ๐‘“๏ท5๏ทฏ Since, L.H.S = R.H.S โˆด ๏ท lim๏ทฎxโ†’5๏ทฏ ๐‘“๏ท๐‘ฅ๏ทฏ= ๐‘“๏ท5๏ทฏ Hence, f is continuous at ๐’™ =๐Ÿ“ Thus the function is continuous at ๐’™ =๐ŸŽ, at ๐’™ =โˆ’๐Ÿ‘ & at ๐’™ =๐Ÿ“

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