Ex 5.1, 3 - Examine for continuity (a) f(x) = x - 5 - Ex 5.1

Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 4 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 5 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 6 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 7 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 8 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 9 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 10 Ex 5.1 ,3 - Chapter 5 Class 12 Continuity and Differentiability - Part 11

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

Transcript

Ex 5.1, 3 Examine the following functions for continuity. (a) f(x) = x – 5 f(x) = x – 5 To check continuity of 𝑓(π‘₯), We check it’s if it is continuous at any point x = c Let c be any real number f is continuous at π‘₯ =𝑐 if (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙)=𝒇(𝒄) (π’π’Šπ’Ž)┬(𝐱→𝒄) 𝒇(𝒙) = lim┬(x→𝑐) π‘₯ βˆ’ 5 = c βˆ’ 5 𝒇(𝒄) = c βˆ’ 5 Since, L.H.S = R.H.S ∴ Function is continuous at x = c Thus, we can write that f is continuous for x = c , where c βˆˆπ‘ ∴ f is continuous for every real number. Ex 5.1, 3 Examine the following functions for continuity. (b) f (x) = 1/(π‘₯ βˆ’ 5) , x β‰  5 f (x) = 1/(π‘₯ βˆ’ 5) At x = 5 f (x) = 1/(5 βˆ’ 5) = 1/0 = ∞ Hence, f(x) is not defined at x = 5 So, we check for continuity at all points except 5 Let c be any real number except 5. f is continuous at π‘₯ = 𝑐 if (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙) = 𝒇(𝒄) LHS (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙) = lim┬(π‘₯βŸΆπ‘)⁑〖1/(π‘₯ βˆ’ 5)γ€— = 1/(𝑐 βˆ’ 5) RHS 𝒇(𝒄) = 1/(𝑐 βˆ’ 5) Since, L.H.S = R.H.S ∴ Function is continuous at x = c (except 5) Thus, we can write that f is continuous for all real numbers except 5 ∴ f is continuous at each 𝐱 ∈ R βˆ’ {πŸ“} 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“} Ex 5.1, 3 Examine the following functions for continuity. (d) f (x) = |x – 5| f(x) = |π‘₯βˆ’5| = {β–ˆ((π‘₯βˆ’5), π‘₯βˆ’5β‰₯0@βˆ’(π‘₯βˆ’5), π‘₯βˆ’5<0)─ = {β–ˆ((π‘₯βˆ’5), π‘₯β‰₯5@βˆ’(π‘₯βˆ’5), π‘₯<5)─ Since we need to find continuity at of the function We check continuity for different values of x When x = 5 When x < 5 When x > 5 Case 1 : When x = 5 f(x) is continuous at π‘₯ = 5 if L.H.L = R.H.L = 𝑓(5) if lim┬(xβ†’5^βˆ’ ) 𝑓(π‘₯)=lim┬(xβ†’5^+ ) " " 𝑓(π‘₯)= 𝑓(5) Since there are two different functions on the left & right of 5, we take LHL & RHL . LHL at x β†’ 5 lim┬(xβ†’5^βˆ’ ) f(x) = lim┬(hβ†’0) f(5 βˆ’ h) = lim┬(hβ†’0) |(5βˆ’β„Ž)βˆ’5| = lim┬(hβ†’0) |βˆ’β„Ž| = lim┬(hβ†’0) β„Ž = 0 RHL at x β†’ 5 lim┬(xβ†’5^+ ) f(x) = lim┬(hβ†’0) f(5 + h) = lim┬(hβ†’0) |(5+β„Ž)βˆ’5| = lim┬(hβ†’0) |β„Ž| = lim┬(hβ†’0) β„Ž = 0 & 𝒇(πŸ“) = |π‘₯βˆ’5| = |5βˆ’5| = 0 Hence, L.H.L = R.H.L = 𝑓(5) ∴ f is continuous at x = 5 Case 2 : When x < 5 For x < 5, f(x) = βˆ’ (x βˆ’ 5) Since this a polynomial It is continuous ∴ f(x) is continuous for x < 5 Case 3 : When x > 5 For x > 5, f(x) = (x βˆ’ 5) Since this a polynomial It is continuous ∴ f(x) is continuous for x > 5 Hence, 𝑓(π‘₯)= |π‘₯βˆ’5| is continuous at all points. i.e. f is continuous at 𝒙 ∈ R.

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
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.