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Chapter 13 Class 11 Limits and Derivatives
Example 3 (i) Important You are here
Ex 12.1, 6 Important
Ex 12.1,10 Important
Ex 12.1, 13
Ex 12.1, 16
Ex 12.1, 22 Important
Ex 12.1, 25 Important
Ex 12.1, 28 Important
Ex 12.1, 30 Important
Ex 12.1, 32 Important
Ex 12.2, 9 (i)
Ex 12.2, 11 (i)
Example 20 (i)
Example 21 (i)
Example 22 (i)
Misc 1 (i)
Misc 6 Important
Misc 9 Important
Misc 24 Important
Misc 27 Important
Misc 28 Important
Misc 30 Important
Chapter 13 Class 11 Limits and Derivatives
Last updated at May 29, 2023 by Teachoo
Example 3 Evaluate: (i) (𝑙𝑖𝑚)┬(𝑥→1) (𝑥 15 − 1)/(𝑥10 − 1) (𝑙𝑖𝑚)┬(𝑥→1) (𝑥 15 − 1)/(𝑥10 − 1) = (〖(1)〗^15 − 1)/(〖(1)〗^10 − 1) = (1 − 1)/(1 − 1) = 0/0 Since it is form 0/0, We can solve by using theorem (𝑙𝑖𝑚)┬(𝑥→𝑎) (𝑥^𝑛 − 𝑎^𝑛)/(𝑥 − 𝑎) = na n – 1 Hence, (𝑙𝑖𝑚)┬(𝑥→1) (𝑥^15 − 1)/(𝑥^10 − 1) = (𝑙𝑖𝑚)┬(𝑥→1) 𝑥^15 – 1 ÷lim┬(x→1) x10 – 1 = (𝑙𝑖𝑚)┬(𝑥→1) 𝑥^15 – 〖(1)〗^15 ÷ lim┬(x→1) x10 – (1)10 Multiplying and dividing by x – 1 = (𝑙𝑖𝑚)┬(𝑥→1) (𝑥^15 − 1^15)/(𝑥 − 1) ÷ (𝑙𝑖𝑚)┬(𝑧→1) (𝑥^10 − 〖(10)〗^10)/(𝑥 − 1) Using (𝑙𝑖𝑚)┬(𝑥→𝑎) ( 𝑥^𝑛 − 𝑎^𝑛)/(𝑥 − 𝑎) = nan – 1 Using (𝑙𝑖𝑚)┬(𝑥→𝑎) ( 𝑥^𝑛 − 𝑎^𝑛)/(𝑥 − 𝑎) = nan – 1 (𝑙𝑖𝑚)┬(𝑥→1) (𝑥^15 − 〖(1)〗^15)/(𝑥 − 1) = 15(1)15 – 1 = 15 (1)14 = 15 (𝑙𝑖𝑚)┬(𝑥→1) (𝑥^10 − 〖(1)〗^10)/(𝑥 − 1) = 10(1)10 – 1 = 10 (1)9 = 10 Hence , (𝑙𝑖𝑚)┬(𝑥→1) (𝑥^15 − 1^15)/(𝑥 − 1) ÷ (𝑙𝑖𝑚)┬(𝑥→1) (𝑥^10 −110)/(𝑥 − 1) = 15 ÷ 10 = 15/10 = 3/2 ∴ (𝒍𝒊𝒎)┬(𝒙→𝟏) (𝒙^𝟏𝟓 − 𝟏)/(𝒙^𝟏𝟎 − 𝟏) = 𝟑/𝟐