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  1. Chapter 13 Class 11 Limits and Derivatives
  2. Serial order wise

Transcript

Example 3 Evaluate: (i) (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯ 15 βˆ’ 1)/(π‘₯10 βˆ’ 1) (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯ 15 βˆ’ 1)/(π‘₯10 βˆ’ 1) = (γ€–(1)γ€—^15 βˆ’ 1)/(γ€–(1)γ€—^10 βˆ’ 1) = (1 βˆ’ 1)/(1 βˆ’ 1) = 0/0 Since it is form 0/0, We can solve by using theorem (π‘™π‘–π‘š)┬(π‘₯β†’π‘Ž) (π‘₯^𝑛 βˆ’ π‘Ž^𝑛)/(π‘₯ βˆ’ π‘Ž) = na n – 1 Hence, (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯^15 βˆ’ 1)/(π‘₯^10 βˆ’ 1) = (π‘™π‘–π‘š)┬(π‘₯β†’1) π‘₯^15 – 1 Γ·lim┬(xβ†’1) x10 – 1 = (π‘™π‘–π‘š)┬(π‘₯β†’1) π‘₯^15 – γ€–(1)γ€—^15 Γ· lim┬(xβ†’1) x10 – (1)10 Multiplying and dividing by x – 1 = (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯^15 βˆ’ 1^15)/(π‘₯ βˆ’ 1) Γ· (π‘™π‘–π‘š)┬(𝑧→1) (π‘₯^10 βˆ’ γ€–(10)γ€—^10)/(π‘₯ βˆ’ 1) Using (π‘™π‘–π‘š)┬(π‘₯β†’π‘Ž) ( π‘₯^𝑛 βˆ’ π‘Ž^𝑛)/(π‘₯ βˆ’ π‘Ž) = nan – 1 Using (π‘™π‘–π‘š)┬(π‘₯β†’π‘Ž) ( π‘₯^𝑛 βˆ’ π‘Ž^𝑛)/(π‘₯ βˆ’ π‘Ž) = nan – 1 (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯^15 βˆ’ γ€–(1)γ€—^15)/(π‘₯ βˆ’ 1) = 15(1)15 – 1 = 15 (1)14 = 15 (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯^10 βˆ’ γ€–(1)γ€—^10)/(π‘₯ βˆ’ 1) = 10(1)10 – 1 = 10 (1)9 = 10 Hence , (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯^15 βˆ’ 1^15)/(π‘₯ βˆ’ 1) Γ· (π‘™π‘–π‘š)┬(π‘₯β†’1) (π‘₯^10 βˆ’110)/(π‘₯ βˆ’ 1) = 15 Γ· 10 = 15/10 = 3/2 ∴ (π’π’Šπ’Ž)┬(π’™β†’πŸ) (𝒙^πŸπŸ“ βˆ’ 𝟏)/(𝒙^𝟏𝟎 βˆ’ 𝟏) = πŸ‘/𝟐 Example 3 Evaluate: (ii) (π‘™π‘–π‘š)┬(π‘₯β†’0) (√(1 + π‘₯) βˆ’ 1)/π‘₯ (π‘™π‘–π‘š)┬(π‘₯β†’0) (√(1 + x )βˆ’ 1)/x Putting x = 0 = (√(1 + 0) βˆ’ 1)/0 = (√(1 ) βˆ’ 1)/0 = (1 βˆ’ 1)/0 = 0/0 Since it is a 0/0 form We simplify the equation Putting y = 1 + x β‡’ y – 1 = x As x β†’ 0 y β†’ 1 + 0 y β†’ 1 So, our equation becomes (π‘™π‘–π‘š)┬(π‘₯β†’0) (√(1 + π‘₯ )βˆ’ 1)/π‘₯ = (π‘™π‘–π‘š)┬(𝑦→1) (βˆšπ‘¦ βˆ’ 1)/(𝑦 βˆ’ 1) = (π‘™π‘–π‘š)┬(π‘₯β†’1) ( 𝑦^((βˆ’1)/2) βˆ’ 1)/(𝑦 βˆ’ 1) = (π‘™π‘–π‘š)┬(π‘₯β†’1) ( 𝑦^((βˆ’1)/2) βˆ’ 1^((βˆ’1)/2))/(𝑦 βˆ’ 1) = 1/2 Γ— 1^((βˆ’1)/2 βˆ’ 1) = 1/2 Γ— 1 = 𝟏/𝟐

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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.