Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Example 3 (i) - Chapter 12 Class 11 Limits and Derivatives
Last updated at May 29, 2023 by Teachoo
Examples
Example 1 (i)
Example 1 (ii)
Example 1 (iii)
Example 2 (i)
Example 2 (ii) Important
Example 2 (iii) Important
Example 2 (iv)
Example 2 (v)
Example 3 (i) Important You are here
Example 3 (ii) Important
Example 4 (i)
Example 4 (ii) Important
Example 5
Example 6
Example 7 Important
Example 8
Example 9
Example 10 Important
Example 11
Example 12
Example 13 Important
Example 14
Example 15 Important
Example 16
Example 17 Important
Example 18
Example 19 (i) Important
Example 19 (ii)
Example 20 (i)
Example 20 (ii) Important
Example 21 (i)
Example 21 (ii) Important
Example 22 (i)
Example 22 (ii) Important
Transcript
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
