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

Transcript

Ex 5.1, 7 Find all points of discontinuity of f, where f is defined by 𝑓(π‘₯)={β–ˆ(|π‘₯|+3, 𝑖𝑓 π‘₯β‰€βˆ’3@ βˆ’2π‘₯, π‘–π‘“βˆ’3<π‘₯<3@ 6π‘₯+2, 𝑖𝑓 π‘₯β‰₯3)─ Since we need to find continuity at of the function We check continuity for different values of x When x < βˆ’3 When x = βˆ’3 When βˆ’3 < x < 3 When x = 3 When x > 3 Case 1 : When x < βˆ’3 For x < βˆ’3, f(x) = |π‘₯|+3 f(x) = βˆ’x + 3 Since this a polynomial It is continuous ∴ f(x) is continuous for x < βˆ’3 (As x < βˆ’3, x is negative) Case 2 : When x = βˆ’3 f(x) is continuous at π‘₯ =βˆ’3 if L.H.L = R.H.L = 𝑓(βˆ’3) if lim┬(xβ†’γ€–βˆ’3γ€—^βˆ’ ) 𝑓(π‘₯)=lim┬(xβ†’γ€–βˆ’3γ€—^+ ) " " 𝑓(π‘₯)= 𝑓(βˆ’3) Since there are two different functions on the left & right of βˆ’3, we take LHL & RHL . LHL at x β†’ βˆ’3 lim┬(xβ†’γ€–βˆ’3γ€—^βˆ’ ) f(x) = lim┬(hβ†’0) f(βˆ’3 βˆ’ h) = lim┬(hβ†’0) (|βˆ’3βˆ’β„Ž|+3) = |βˆ’3βˆ’0|+3 = |βˆ’3|+3 = 3 + 3 = 6 RHL at x β†’ βˆ’3 lim┬(xβ†’γ€–βˆ’3γ€—^+ ) f(x) = lim┬(hβ†’0) f(βˆ’3 + h) = lim┬(hβ†’0) βˆ’2(βˆ’3+β„Ž) = lim┬(hβ†’0) 6βˆ’2β„Ž = 6 βˆ’ 0 = 6 & 𝑓(βˆ’3) = |βˆ’3|+3 = 3+3 = 6 Hence, L.H.L = R.H.L = 𝑓(βˆ’3) ∴ f is continuous at x=βˆ’3 Case 3 : When βˆ’3 < x < 3 For βˆ’3 < x < 3, f(x) = βˆ’2x Since this a polynomial It is continuous ∴ f(x) is continuous for βˆ’3 < x < 3 Case 4 : When x = 3 f(x) is continuous at π‘₯ =3 if L.H.L = R.H.L = 𝑓(3) if lim┬(xβ†’3^βˆ’ ) 𝑓(π‘₯)=lim┬(xβ†’3^+ ) " " 𝑓(π‘₯)= 𝑓(3) Since there are two different functions on the left & right of 3, we take LHL & RHL . LHL at x β†’ 3 lim┬(xβ†’3^βˆ’ ) f(x) = lim┬(hβ†’0) f(3 βˆ’ h) = lim┬(hβ†’0) βˆ’2(3βˆ’h) = lim┬(hβ†’0) βˆ’6+2h = βˆ’6 + 0 = βˆ’6 Since L.H.L β‰  R.H.L f(x) is not continuous at x=3 Case 5: When x > 3 For x > 3, f(x) = 6x + 2 Since this a polynomial It is continuous ∴ f(x) is continuous for x > 3 Hence, f is discontinuous at only π‘₯ =3 Thus, f is continuous at all real numbers except 3. f is continuous at x ∈Rβˆ’{3}

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.