Chapter 5 Class 12 Continuity and Differentiability

Class 12
Important Questions for exams Class 12

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Ex 5.1, 16 Discuss the continuity of the function f, where f is defined by π(π₯)={β(β&2, ππ π₯β€β[email protected]&π₯, ππ β1<π₯β€[email protected], ππ π₯>1 )β€ Since we need to find continuity at of the function We check continuity for different values of x When x < β1 When x = β1 When β1 < x < 1 When x = 1 When x > 1 Case 1 : When x < β1 For x < β1, f(x) = β2 Since this constant It is continuous β΄ f(x) is continuous for x < β1 Case 2 : When x = β1 f(x) is continuous at π₯ =β1 if L.H.L = R.H.L = π(β1) if limβ¬(xβγβ1γ^β ) π(π₯)=limβ¬(xβγβ1γ^+ ) " " π(π₯)= π(β1) Since there are two different functions on the left & right of β1, we take LHL & RHL . LHL at x β β1 limβ¬(xβγβ1γ^β ) f(x) = limβ¬(hβ0) f(β1 β h) = limβ¬(hβ0) β2 = β2 RHL at x β 0 limβ¬(xβγβ1γ^+ ) f(x) = limβ¬(hβ0) f(β1 + h) = limβ¬(hβ0) 2(β1+β) = 2(β1+0) = β2 & π(β1) = β2 Hence, L.H.L = R.H.L = π(β1) β΄ f is continuous at x = β1 Case 3 : When β1 < x < 1 For β1 < x < 1 f(x) = 2x Since this a polynomial It is continuous β΄ f(x) is continuous for β1 < x < 1 Case 4 : When x = 1 f(x) is continuous at π₯ =1 if L.H.L = R.H.L = π(1) if limβ¬(xβ1^β ) π(π₯)=limβ¬(xβ1^+ ) " " π(π₯)= π(1) & π(1) = 2(1) = 2 Since there are two different functions on the left & right of 1, we take LHL & RHL . LHL at x β 1 limβ¬(xβ1^β ) f(x) = limβ¬(hβ0) f(1 β h) = limβ¬(hβ0) 2(1ββ) = 2(1β0) = 2 RHL at x β 1 limβ¬(xβ1^+ ) f(x) = limβ¬(hβ0) f(1 + h) = limβ¬(hβ0) 2 = 2 Hence, L.H.L = R.H.L = π(1) β΄ f is continuous at x=1 Case 5: When x > 1 For x > 1, f(x) = 2 Since this constant It is continuous β΄ f(x) is continuous for x > 1 Thus, f(x) is continuous for all real numbers, i.e. f is continuous for all x β R

#### 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.