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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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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 > 1Case 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 of discontinuity. ∴ f is continuous at all real numbers except 1 Thus, f is continuous for π’™βˆˆ R βˆ’ {1}

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.