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CHAPTER 2 : LIMITS

2.1 Modulus

| a - b | means the "non-negative difference between a and b and is read as mod of a minus b".

\ | a - b | = a - b if a > b

      = 0 if a = b

      = b - a if a < b

e.g. | 7 - 5 | = 2, | 3 - 8 | = 5, | 8 | = 8, | -9 | = | 0 - 9 | = 9


Examples

  1. If | x | = 8 then x = +8 or x = -8

  2. If | x - 9 | = 3 then x- 9 = +3 or x- 9 = -3

    i.e. x = 12 or x = 6

  3. If | x - a | = d then x - a = -d or x - a = +d

    i.e. x = a - d or x = a + d

  4. If | 5x - 2 | = | x | then 5x-2 = x or 5x - 2 = -x

    i.e. 5x - x = 2 or 5x + x = 2

    i.e. 4x = 2 or 6x = 2

    i.e. x = 1/2 or x = 1/3

  5. If 2 | x | = | x - 3 | then 2 x = ( x-3 ) Or 2x = -( x- 3 )

    i.e. 2x - x = -3 or 2x + x = 3

    i.e. x = -3 or 3x = 3

    i.e. x = -3 or x = 1

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Index

2.1 Modulus
2.2 Inequalities
2.3 Limits Of Functions
2.4 Left Hand And Right Hand Limits
2.5 Theorems On The Algebra Of Limits
2.6 Evaluating Limits
2.7 Limits Of Trigonometric Functions
2.8 The Exponential Limits

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Chapter 3




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