Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 5

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Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 6

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Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 7

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

Transcript

Ex 5.1, 3 Examine the following functions for continuity. (c) f (x) = (π‘₯^(2 )βˆ’ 25 )/(π‘₯ + 5), x β‰  –5 f (x) = (π‘₯^(2 )βˆ’ 25 )/(π‘₯ + 5) Putting x = –5 f (βˆ’5) = (γ€–(βˆ’5)γ€—^(2 )βˆ’ 25 )/(βˆ’5 + 5) = (25βˆ’ 25 )/(βˆ’5 + 5) = 0/0 = Undefined Hence, f(x) is not defined at x = βˆ’5 So, we check for continuity at all points except βˆ’5 Let c be any real number except βˆ’5. f is continuous at π‘₯ = 𝑐 if (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙) = 𝒇(𝒄) LHS (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙) = lim┬(x→𝑐) (π‘₯^2βˆ’ 25)/(π‘₯ + 5) = lim┬(x→𝑐) ((π‘₯ βˆ’ 5) (π‘₯ + 5))/(π‘₯ + 5) = lim┬(x→𝑐) π‘₯βˆ’5 Putting x = c = c βˆ’ 5 RHS f (c) = (𝑐^(2 )βˆ’ 25 )/(𝑐 + 5) = ((𝑐 βˆ’ 5)(𝑐 + 5))/((𝑐 + 5)) = c βˆ’ 5 Let c be any real number except βˆ’5. f is continuous at π‘₯ = 𝑐 if (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙) = 𝒇(𝒄) Since, L.H.S = R.H.S ∴ Function is continuous at x = c (except βˆ’5) Thus, we can write that f is continuous for all real numbers except βˆ’5 ∴ f is continuous at each 𝐱 ∈ R βˆ’ {βˆ’πŸ“}

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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.