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

Transcript

Example 10 Discuss the continuity of the function f defined by 𝑓(π‘₯)={β–ˆ(&π‘₯+2, 𝑖𝑓 π‘₯≀1@&π‘₯βˆ’2, 𝑖𝑓 π‘₯>1)─ 𝑓(π‘₯)={β–ˆ(&π‘₯+2, 𝑖𝑓 π‘₯≀1@&π‘₯βˆ’2, 𝑖𝑓 π‘₯>1)─ Since we need to find continuity at of the function We check continuity for different values of x When x = 1 When x < 1 When x > 1 Case 1 : When x = 1 f(x) is continuous at π‘₯ =1 if L.H.L = R.H.L = 𝑓(1) if lim┬(xβ†’1^βˆ’ ) 𝑓(π‘₯)=lim┬(xβ†’1^+ ) " " 𝑓(π‘₯)= 𝑓(1) Since there are two different functions on the left & right of 1, we take LHL & RHL . LHL at x β†’ 1 lim┬(xβ†’1^βˆ’ ) f(x) = lim┬(hβ†’0) f(1 βˆ’ h) = lim┬(hβ†’0) (1βˆ’β„Ž)+2 = lim┬(hβ†’0) (3βˆ’β„Ž) = 3 βˆ’ 0 = 3 RHL at x β†’ 1 lim┬(xβ†’1^+ ) f(x) = lim┬(hβ†’0) f(1 + h) = lim┬(hβ†’0) (1+β„Ž)βˆ’2 = lim┬(hβ†’0) (βˆ’1+β„Ž) = βˆ’1 + 0 = βˆ’1 Since L.H.L β‰  R.H.L f(x) is not continuous at x=1 Case 2 : When x < 1 For x < 1, f(x) = x + 2 Since this a polynomial It is continuous ∴ f(x) is continuous for x < 1 Case 3 : When x > 1 For x > 1, f(x) = x βˆ’ 2 Since this a polynomial It is continuous ∴ f(x) is continuous for x > 1 Hence, only π‘₯=1 is point is discontinuity. f is continuous at all real numbers except 1 Thus, f is continuous for 𝐱 ∈ R βˆ’ {1}

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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.