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Ex 13.1, 24 - Find lim x-> 1, where f(x) = { x^2 - 1, x <= 1, -x^2 - 1

Ex 13.1, 24 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Ex 13.1, 24 - Chapter 13 Class 11 Limits and Derivatives - Part 3
Ex 13.1, 24 - Chapter 13 Class 11 Limits and Derivatives - Part 4

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Transcript

Ex 13.1, 24 (Method 1) Find (π‘™π‘–π‘š)┬(π‘₯β†’1) f(x), where f(x) = {β–ˆ(π‘₯2 βˆ’1,@βˆ’π‘₯2 βˆ’1,)─ β– 8(π‘₯ ≀1@π‘₯>1) The Limit at x = 1 will be (π‘™π‘–π‘š)┬(π‘₯β†’1) f(x) = lim┬(γ€–xβ†’1γ€—^βˆ’ ) f(x) =(π‘™π‘–π‘š)┬(γ€–π‘₯β†’1γ€—^+ ) f(x) (π’π’Šπ’Ž)┬(γ€–π’™β†’πŸγ€—^βˆ’ ) f(x) = (π‘™π‘–π‘š)┬(π‘₯β†’1) x2 – 1 = (1)2 – 1 = 1 – 1 = 0 (π’π’Šπ’Ž)┬(γ€–π’™β†’πŸγ€—^+ ) f(x) = (π‘™π‘–π‘š)┬(π‘₯β†’1) (–x2 – 1) = –(1)2 – 1 = –1 – 1 = –2 Thus, (π’π’Šπ’Ž)┬(γ€–π’™β†’πŸγ€—^+ )f(x) β‰  (π’π’Šπ’Ž)┬(γ€–π’™β†’πŸγ€—^βˆ’ )f(x) Since, Left Hand Limit & Right Hand Limit are not equal Hence (π₯π’Šπ’Ž)┬(π’™β†’πŸ) f(x) does not exit Ex 13.1, 24 (Method 2) Find (π‘™π‘–π‘š)┬(π‘₯β†’1) f(x), where f(x) = {β–ˆ(π‘₯2 βˆ’1,@βˆ’π‘₯2 βˆ’1,)─ β– 8(π‘₯ ≀1@π‘₯>1) The Limit at x = 1 will be (π‘™π‘–π‘š)┬(π‘₯β†’1) f(x) = lim┬(γ€–xβ†’1γ€—^βˆ’ ) f(x) =(π‘™π‘–π‘š)┬(γ€–π‘₯β†’1γ€—^+ ) f(x) LHL at x β†’ 1 lim┬(xβ†’1^βˆ’ ) f(x) = lim┬(hβ†’0) f(1 βˆ’ h) = lim┬(hβ†’0) (1 βˆ’ h)2 βˆ’1 = (1 βˆ’ 0)2 βˆ’ 1 = (1)2 βˆ’ 1 = 1 βˆ’ 1 = 0 RHL at x β†’ 1 lim┬(xβ†’1^+ ) f(x) = lim┬(hβ†’0) f(1 + h) = lim┬(hβ†’0) βˆ’(1 + h)2 βˆ’ 1 = βˆ’(1 + 0)2 βˆ’ 1 = βˆ’(1)2 βˆ’ 1 = βˆ’1 βˆ’ 1 = βˆ’2 Since LHL β‰  RHL ∴ (π’π’Šπ’Ž)┬(π’™β†’πŸ) f(x) doesn’t exist

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.