Limits and Continuity - Intuitive Approach part 3
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The document discusses the concept of continuity in mathematical functions, defining continuity at a point and outlining properties such as the sum, difference, product, and quotient of continuous functions. Several illustrations are provided to elaborate on the application of these properties, including an example of checking continuity for a specific function. The content serves as a foundational understanding of limits and continuity for quantitative aptitude in mathematics.
Limits and Continuity - Intuitive Approach part 3
- 1. Limits and Continuity – Intutive Approach– Chapter 8 Paper 4: Quantitative Aptitude- Mathematics Ms. Ritu Gupta B.A. (Hons.) Maths and MA (Maths)
- 2. Continuity • Fundamental Knowledge • Its application 2
- 3. Concept of Continuity A function is said to be continuous at a point x = a if (i) f(x) is defined at x = a (ii) (iii) i.e. L.H.L = R.H.L = value of the function at x = a 3
- 5. Properties of Continuous Functions (i) If f(x) and g(x) are both continuous at a point x=a, then f(x)+g(x) is also continuous. (ii) If f(x) and g(x) are both continuous at a point x=a then f(x) – g(x) is also continuous. (iii) If f(x) and g(x) are both continuous at a point x=a then their product f(x) . g(x) is also continuous at x = a. (iv) If f(x) and g(x) are both continuous at a point x=a then their quotient f(x)/ g(x) is also continuous at x=a and g(a) ≠ 0. 5
- 6. Properties of Continuous Functions- Continued 6
- 9. Illustration 24 - Continued 9
- 10. Illustration 24 - Continued 10
- 12. Illustration 25 – Continued 12
- 14. Illustration 26 – Continued 14
- 16. Illustration 27 – Continued 16
- 18. Illustration 28 – Continued 18
- 20. Illustration 29 – Continued 20
- 21. Illustration 29 – Continued 21
- 22. Illustration 29 – Continued 22
- 23. Illustration 30 A function f(x) is defined as follows:- f(x) = x2 when 0<x<1 = x when 1≤x<2 = (¼) x3 when 2≤x<3 Now f(x) is continuous at (a) x = 1 (b) x = 3 (c) x=0 (d) None of these Solution: Let us first check the continuity of a function f(x), at x = 1 23
- 24. Illustration 30 - Continued 24
- 26. Illustration 31 - Continued 26
- 27. Illustration 31 - Continued 27
- 29. Illustration 32 – Continued 29
- 31. Illustration 33 – Continued 31
- 33. Illustration 34 – Continued 33
- 34. Illustration 34 – Continued
- 36. Illustration 35 – Continued 36
- 37. Illustration 35 – Continued 37
- 39. Illustration 36 – Continued 39
- 40. Illustration 36 – Continued 40
- 42. Illustration 37 – Continued 42
- 43. Thank you 43