11X1 T09 01 limits and continuity (2010)

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  • 1. Limits & Continuity

2. Limits & Continuity A limit describes the behaviour of functions. 3. Limits & Continuity A limit describes the behaviour of functions. lim f x : x a 4. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach? x a 5. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach? x alim f x :x a 6. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach? x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach? 7. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach? x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach?yy x 1 11x 8. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach? yy x 11 1 xlim x 1 0 x1 9. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach? y yy x 1 6 4 y f x 1 1 x xlim x 1 0 x1 10. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach? y yy x 1 6 4 y f x 1 1 x xlim x 1 0 lim f x 4 x1 x0 11. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach?lim f x :x ay yy x 1 6 4 y f x 1 1 x xlim x 1 0 lim f x 4 x1 x0 12. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach?lim f x : as the x value approaches a from the positive side,x a what value does f(x) approach? y yy x 1 6 4 y f x 1 1 x xlim x 1 0 lim f x 4 x1 x0 13. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach?lim f x : as the x value approaches a from the positive side,x a what value does f(x) approach? y yy x 1 6 4 y f x 1 1 x xlim x 1 0 lim x 1 0 lim f x 4 x1 x1 x0 14. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach?lim f x : as the x value approaches a from the positive side,x a what value does f(x) approach? y yy x 1 6 4 y f x 1 1 x xlim x 1 0 lim x 1 0 lim f x 4 lim f x 6 x1 x1 x0 x0 15. Limits & Continuity A limit describes the behaviour of functions. lim f x : as the x value approaches a, what value does f(x) approach?x alim f x : as the x value approaches a from the negative side,x a what value does f(x) approach?lim f x : as the x value approaches a from the positive side,x a what value does f(x) approach? yyy x 1 6 4y f x 1 1x xlim x 1 0 lim x 1 0lim f x 4 lim f x 6 x1 x1x0 x0 If lim f x lim f x , then f x is continuous at x ax a x a 16. Finding Limits 17. Finding Limits (1) Direct Substitution 18. Finding Limits (1) Direct Substitutione.g. lim x 7 x5 19. Finding Limits (1) Direct Substitutione.g. lim x 7 5 7 x5 12 20. Finding Limits (1) Direct Substitutione.g. lim x 7 5 7 x5 12(2) Factorise and Cancel 21. Finding Limits (1) Direct Substitutione.g. lim x 7 5 7 x5 12(2) Factorise and Cancelx2 9 e.g. limx3 x 3 22. Finding Limits (1) Direct Substitution e.g. lim x 7 5 7 x5 12(2) Factorise and Cancele.g. limx2 9 lim x 3 x 3x3 x 3x3 x 3 23. Finding Limits (1) Direct Substitution e.g. lim x 7 5 7 x5 12(2) Factorise and Cancele.g. limx2 9 lim x 3 x 3x3 x 3x3 x 3 lim x 3 x3 24. Finding Limits (1) Direct Substitution e.g. lim x 7 5 7 x5 12(2) Factorise and Cancele.g. limx2 9 lim x 3 x 3x3 x 3x3 x 3 lim x 3 x3 336 25. (3) Special Limit 26. (3) Special Limit 1 lim 0 x x 27. (3) Special Limit1lim 0x xx3 3x 2 2 x 1 e.g. i limx4 x3 1 28. (3) Special Limit1lim 0x xx3 3x 2 2 x 1x3 3x 2 2 x 1 3 3 3 e.g. i 3lim lim x xx xx4x 13x4 x3 13 3xx 29. (3) Special Limit 1 lim 0 x x x3 3x 2 2 x 1x3 3x 2 2 x 1 3 3 3 e.g. i 3lim lim x x x xx4x 13 x4 x3 1 3 3 xx3 2 11 2 3 lim x x xx1 4 3x 30. (3) Special Limit 1 lim 0 x x x3 3x 2 2 x 1x3 3x 2 2 x 1 3 3 3 e.g. i 3lim lim x x x xx4x 13 x4 x3 1 3 3 xx3 2 11 2 3 lim x x xx1 4 3x 1 4 31. (3) Special Limit1lim 0x xx3 3x 2 2 x 1 x3 3x 2 2 x 1 3 3 3 e.g. i 3 lim lim x x x x x4x 1 3x4 x3 13 3xx 3 2 1 1 2 3 lim x x x x14 3 x1 4 4x x2 ii lim 3 x x 1 32. (3) Special Limit1lim 0x xx3 3x 2 2 x 1 x3 3x 2 2 x 1 3 3 3 e.g. i 3 lim lim x x x x x4x 1 3x4 x3 13 3xx 3 2 1 1 2 3 lim x x x x14 3 x1 4 4x x20 ii lim 3 x x 110 33. (3) Special Limit 1 lim 0 x xx3 3x 2 2 x 1 x3 3x 2 2 x 1 3 3 3 e.g. i 3 lim lim x x x x x4x 1 3x4 x3 13 3xx3 2 1 1 2 3 lim x x x x1 4 3 x1 4 4x x20x7 x6 x2 ii lim 3 x x 1 iii lim 7x 3 x x 97410 34. (3) Special Limit 1 lim 0 x xx3 3x 2 2 x 1 x3 3x 2 2 x 1 3 3 3 e.g. i 3 lim lim x x x x x4x 1 3x4 x3 13 3xx3 2 1 1 2 3 lim x x x x1 4 3 x1 4 4x x20x7 x6 x2 1 ii lim 3 x x 1 iii lim 7x 3 x x 9741 30 35. x3 2 iv lim 2x x 1 36. x3 2 1 iv lim 2x x 1 0 37. x3 2 1 iv lim 2x x 1 0 x 3 x 2 v Find the horizontal asymptote of y x 1 x 1 38. x3 2 1 iv lim 2x x 1 0 x 3 x 2 v Find the horizontal asymptote of y x 1 x 1 lim x 3 x 2 x x 1 x 1 39. x3 2 1 iv lim 2x x 1 0 x 3 x 2 v Find the horizontal asymptote of y x 1 x 1 x 3 x 2 x2 x 6 lim lim 2 x x 1 x 1x x 1 40. x3 2 1 iv lim 2x x 1 0 x 3 x 2 v Find the horizontal asymptote of y x 1 x 1 x 3 x 2 x2 x 6 lim lim 2 x x 1 x 1x x 1111 41. x3 2 1 iv lim 2x x 1 0 x 3 x 2 v Find the horizontal asymptote of y x 1 x 1 x 3 x 2 x2 x 6 lim lim 2 x x 1 x 1x x 1 1 1 1 horizontal asymptote is y 1 42. x3 2 1 iv lim 2x x 1 0 x 3 x 2 v Find the horizontal asymptote of y x 1 x 1 x 3 x 2 x2 x 6 lim lim 2 x x 1 x 1x x 1 1 1 1 horizontal asymptote is y 1 Exercise 7I; 1a, 2ace, 3ac, 4a, 5ad, 8a, 9ab, 10a