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Page 1 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Exercise 2 1 .1 Question 1: Evaluate the Given limit: x3 lim 3 x ? ? Solution 1: x3 lim 3 x ? ? = 3 + 3 = 6 Question 2: Evaluate the Given limit: x 22 lim 7 x ? ? ?? ? ?? ?? Solution 2: x 22 lim 7 x ? ? ?? ? ?? ?? = 22 7 ? ?? ? ?? ?? Question 3: Evaluate the Given limit : 2 x1 lim r ? ? Solution 3: 2 x1 lim r ? ? = ? ? 2 1 ?? ? Question 4: Evaluate the Given limit: x1 43 lim 2 x x ? ? ? Solution 4: x1 43 lim 2 x x ? ? ? = ? ? 4 4 3 16 3 19 4 2 2 2 ? ? ?? ? Page 2 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Exercise 2 1 .1 Question 1: Evaluate the Given limit: x3 lim 3 x ? ? Solution 1: x3 lim 3 x ? ? = 3 + 3 = 6 Question 2: Evaluate the Given limit: x 22 lim 7 x ? ? ?? ? ?? ?? Solution 2: x 22 lim 7 x ? ? ?? ? ?? ?? = 22 7 ? ?? ? ?? ?? Question 3: Evaluate the Given limit : 2 x1 lim r ? ? Solution 3: 2 x1 lim r ? ? = ? ? 2 1 ?? ? Question 4: Evaluate the Given limit: x1 43 lim 2 x x ? ? ? Solution 4: x1 43 lim 2 x x ? ? ? = ? ? 4 4 3 16 3 19 4 2 2 2 ? ? ?? ? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 5: Evaluate the Given limit: 10 5 1 1 lim 1 x xx x ?? ?? ? Solution 5: ? ? ? ? 10 5 10 5 1 1 1 1 1 1 1 1 1 lim 1 1 1 2 2 x xx x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 6: Evaluate the Given limit: ? ? 5 0 11 lim x x x ? ?? Solution 6: ? ? 5 0 11 lim x x x ? ?? Put x + 1 = y so that y → 1 as x → 0. Accordingly, ? ? 5 0 11 lim x x x ? ?? = 5 1 1 lim 1 x y y ? ? ? ? ? 55 1 5 1 1 5 0 1 lim 1 5.1 lim 5 11 lim 5 x nn n xa x y y xa na xa x x ? ?? ? ? ? ? ? ?? ? ?? ?? ? ?? ? ?? ?? Question 7: Evaluate the Given limit: 2 2 2 3 10 lim 4 x xx x ? ?? ? Solution 7: At x = 2, the value of the given rational function takes the form 0 0 . ? 2 2 2 3 10 lim 4 x xx x ? ?? ? = ? ? ? ? ? ? ? ? 2 2 3 5 lim 22 x xx xx ? ?? ? ?? Page 3 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Exercise 2 1 .1 Question 1: Evaluate the Given limit: x3 lim 3 x ? ? Solution 1: x3 lim 3 x ? ? = 3 + 3 = 6 Question 2: Evaluate the Given limit: x 22 lim 7 x ? ? ?? ? ?? ?? Solution 2: x 22 lim 7 x ? ? ?? ? ?? ?? = 22 7 ? ?? ? ?? ?? Question 3: Evaluate the Given limit : 2 x1 lim r ? ? Solution 3: 2 x1 lim r ? ? = ? ? 2 1 ?? ? Question 4: Evaluate the Given limit: x1 43 lim 2 x x ? ? ? Solution 4: x1 43 lim 2 x x ? ? ? = ? ? 4 4 3 16 3 19 4 2 2 2 ? ? ?? ? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 5: Evaluate the Given limit: 10 5 1 1 lim 1 x xx x ?? ?? ? Solution 5: ? ? ? ? 10 5 10 5 1 1 1 1 1 1 1 1 1 lim 1 1 1 2 2 x xx x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 6: Evaluate the Given limit: ? ? 5 0 11 lim x x x ? ?? Solution 6: ? ? 5 0 11 lim x x x ? ?? Put x + 1 = y so that y → 1 as x → 0. Accordingly, ? ? 5 0 11 lim x x x ? ?? = 5 1 1 lim 1 x y y ? ? ? ? ? 55 1 5 1 1 5 0 1 lim 1 5.1 lim 5 11 lim 5 x nn n xa x y y xa na xa x x ? ?? ? ? ? ? ? ?? ? ?? ?? ? ?? ? ?? ?? Question 7: Evaluate the Given limit: 2 2 2 3 10 lim 4 x xx x ? ?? ? Solution 7: At x = 2, the value of the given rational function takes the form 0 0 . ? 2 2 2 3 10 lim 4 x xx x ? ?? ? = ? ? ? ? ? ? ? ? 2 2 3 5 lim 22 x xx xx ? ?? ? ?? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? 2 35 lim 2 3 2 5 22 11 4 x x x ? ? ? ? ? ? ? ? Question 8: Evaluate the Given limit: 4 2 3 81 lim 2 5 3 x x xx ? ? ?? Solution 8: At x = 2, the value of the given rational function takes the form 0 0 . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 4 2 33 2 3 2 3 3 9 81 lim lim 2 5 3 3 2 1 39 lim 21 3 3 3 9 2 3 1 6 18 7 108 7 xx x x x x x x x x x xx x ?? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? Question 9: Evaluate the Given limit: 0 lim 1 x ax b cx ? ? ? Solution 9: 0 lim 1 x ax b cx ? ? ? = ? ? ? ? 0 01 ab b c ? ? ? Page 4 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Exercise 2 1 .1 Question 1: Evaluate the Given limit: x3 lim 3 x ? ? Solution 1: x3 lim 3 x ? ? = 3 + 3 = 6 Question 2: Evaluate the Given limit: x 22 lim 7 x ? ? ?? ? ?? ?? Solution 2: x 22 lim 7 x ? ? ?? ? ?? ?? = 22 7 ? ?? ? ?? ?? Question 3: Evaluate the Given limit : 2 x1 lim r ? ? Solution 3: 2 x1 lim r ? ? = ? ? 2 1 ?? ? Question 4: Evaluate the Given limit: x1 43 lim 2 x x ? ? ? Solution 4: x1 43 lim 2 x x ? ? ? = ? ? 4 4 3 16 3 19 4 2 2 2 ? ? ?? ? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 5: Evaluate the Given limit: 10 5 1 1 lim 1 x xx x ?? ?? ? Solution 5: ? ? ? ? 10 5 10 5 1 1 1 1 1 1 1 1 1 lim 1 1 1 2 2 x xx x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 6: Evaluate the Given limit: ? ? 5 0 11 lim x x x ? ?? Solution 6: ? ? 5 0 11 lim x x x ? ?? Put x + 1 = y so that y → 1 as x → 0. Accordingly, ? ? 5 0 11 lim x x x ? ?? = 5 1 1 lim 1 x y y ? ? ? ? ? 55 1 5 1 1 5 0 1 lim 1 5.1 lim 5 11 lim 5 x nn n xa x y y xa na xa x x ? ?? ? ? ? ? ? ?? ? ?? ?? ? ?? ? ?? ?? Question 7: Evaluate the Given limit: 2 2 2 3 10 lim 4 x xx x ? ?? ? Solution 7: At x = 2, the value of the given rational function takes the form 0 0 . ? 2 2 2 3 10 lim 4 x xx x ? ?? ? = ? ? ? ? ? ? ? ? 2 2 3 5 lim 22 x xx xx ? ?? ? ?? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? 2 35 lim 2 3 2 5 22 11 4 x x x ? ? ? ? ? ? ? ? Question 8: Evaluate the Given limit: 4 2 3 81 lim 2 5 3 x x xx ? ? ?? Solution 8: At x = 2, the value of the given rational function takes the form 0 0 . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 4 2 33 2 3 2 3 3 9 81 lim lim 2 5 3 3 2 1 39 lim 21 3 3 3 9 2 3 1 6 18 7 108 7 xx x x x x x x x x x xx x ?? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? Question 9: Evaluate the Given limit: 0 lim 1 x ax b cx ? ? ? Solution 9: 0 lim 1 x ax b cx ? ? ? = ? ? ? ? 0 01 ab b c ? ? ? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 10: Evaluate the Given limit: 1 3 1 1 6 1 lim 1 z z z ? ? ? Solution 10: 1 3 1 1 6 1 lim 1 z z z ? ? ? At z = 1, the value of the given function takes the form 0 0 . Put 1 6 z = x so that z →1 as x → 1. Accordingly, 1 3 1 1 6 1 lim 1 z z z ? ? ? = 2 1 1 lim 1 x x x ? ? ? 2 1 2 1 1 1 lim 1 2.1 lim 2 x nn n xa x x xa na xa ? ?? ? ? ? ? ?? ? ?? ?? ? ?? ? 1 3 1 1 6 1 lim 1 z z z ? ? ? = 2 Question 11: Evaluate the Given limit: ? ? 2 2 1 lim 1 x ax bx c cx b a ? ?? ?? , a + b + c ? 0 Solution 11: 2 2 1 lim x ax bx c cx bx a ? ?? ?? = ? ? ? ? ? ? ? ? 2 2 11 11 a b c c b a ?? ?? = abc abc ?? ?? = 1 [a + b + c ? 0] Page 5 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Exercise 2 1 .1 Question 1: Evaluate the Given limit: x3 lim 3 x ? ? Solution 1: x3 lim 3 x ? ? = 3 + 3 = 6 Question 2: Evaluate the Given limit: x 22 lim 7 x ? ? ?? ? ?? ?? Solution 2: x 22 lim 7 x ? ? ?? ? ?? ?? = 22 7 ? ?? ? ?? ?? Question 3: Evaluate the Given limit : 2 x1 lim r ? ? Solution 3: 2 x1 lim r ? ? = ? ? 2 1 ?? ? Question 4: Evaluate the Given limit: x1 43 lim 2 x x ? ? ? Solution 4: x1 43 lim 2 x x ? ? ? = ? ? 4 4 3 16 3 19 4 2 2 2 ? ? ?? ? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 5: Evaluate the Given limit: 10 5 1 1 lim 1 x xx x ?? ?? ? Solution 5: ? ? ? ? 10 5 10 5 1 1 1 1 1 1 1 1 1 lim 1 1 1 2 2 x xx x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 6: Evaluate the Given limit: ? ? 5 0 11 lim x x x ? ?? Solution 6: ? ? 5 0 11 lim x x x ? ?? Put x + 1 = y so that y → 1 as x → 0. Accordingly, ? ? 5 0 11 lim x x x ? ?? = 5 1 1 lim 1 x y y ? ? ? ? ? 55 1 5 1 1 5 0 1 lim 1 5.1 lim 5 11 lim 5 x nn n xa x y y xa na xa x x ? ?? ? ? ? ? ? ?? ? ?? ?? ? ?? ? ?? ?? Question 7: Evaluate the Given limit: 2 2 2 3 10 lim 4 x xx x ? ?? ? Solution 7: At x = 2, the value of the given rational function takes the form 0 0 . ? 2 2 2 3 10 lim 4 x xx x ? ?? ? = ? ? ? ? ? ? ? ? 2 2 3 5 lim 22 x xx xx ? ?? ? ?? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? 2 35 lim 2 3 2 5 22 11 4 x x x ? ? ? ? ? ? ? ? Question 8: Evaluate the Given limit: 4 2 3 81 lim 2 5 3 x x xx ? ? ?? Solution 8: At x = 2, the value of the given rational function takes the form 0 0 . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 4 2 33 2 3 2 3 3 9 81 lim lim 2 5 3 3 2 1 39 lim 21 3 3 3 9 2 3 1 6 18 7 108 7 xx x x x x x x x x x xx x ?? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? Question 9: Evaluate the Given limit: 0 lim 1 x ax b cx ? ? ? Solution 9: 0 lim 1 x ax b cx ? ? ? = ? ? ? ? 0 01 ab b c ? ? ? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 10: Evaluate the Given limit: 1 3 1 1 6 1 lim 1 z z z ? ? ? Solution 10: 1 3 1 1 6 1 lim 1 z z z ? ? ? At z = 1, the value of the given function takes the form 0 0 . Put 1 6 z = x so that z →1 as x → 1. Accordingly, 1 3 1 1 6 1 lim 1 z z z ? ? ? = 2 1 1 lim 1 x x x ? ? ? 2 1 2 1 1 1 lim 1 2.1 lim 2 x nn n xa x x xa na xa ? ?? ? ? ? ? ?? ? ?? ?? ? ?? ? 1 3 1 1 6 1 lim 1 z z z ? ? ? = 2 Question 11: Evaluate the Given limit: ? ? 2 2 1 lim 1 x ax bx c cx b a ? ?? ?? , a + b + c ? 0 Solution 11: 2 2 1 lim x ax bx c cx bx a ? ?? ?? = ? ? ? ? ? ? ? ? 2 2 11 11 a b c c b a ?? ?? = abc abc ?? ?? = 1 [a + b + c ? 0] Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 12: Evaluate the Given limit: 2 11 2 lim 2 x x x ?? ? ? Solution 12: 2 11 2 lim 2 x x x ?? ? ? At x = –2, the value of the given function takes the form 0 0 Now, 2 11 2 lim 2 x x x ?? ? ? = 2 2 2 lim 2 x x x x ?? ? ?? ?? ?? ? = 2 1 lim 2 x x ?? ? ? 11 2 2 4 ? ?? ? Question 13: Evaluate the Given limit: 0 sin lim x ax bx ? Solution 13: 0 sin lim x ax bx ? At x = 0, the value of the given function takes the form 0 0 . Now, 0 sin lim x ax bx ? = 0 sin lim x ax ax ax bx ? ? 0 sin lim x ax a ax b ? ?? ?? ?? ?? ? ? 0 sin lim 0 0 ax a ax x ax b ax ? ?? ? ? ? ? ?? ?? 0 sin 1 lim 1 y ay by a b ? ?? ? ? ? ?? ?? ?Read More
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1. What is the concept of limits in calculus? |
2. How are limits and derivatives related in calculus? |
3. What is the significance of limits and derivatives in the field of mathematics? |
4. How are limits and derivatives applied in real-life scenarios? |
5. What are some common techniques used to evaluate limits and derivatives? |
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