Contents 1.1 Limit of Sequence 1.1.1 Arithmetic Rule 9 1.1.2 Sandwich Rule 1.1.3 Some Basic Limits 19 1.2 Rigorous Definition of Sequence Limit 1.2.1 Rigorous Definition 1.2.2 The art of estimation 1.2.3 Rigorous Proof of limits 31 1. 2. 4 Rigorous Proof of Limit Propertie 1.3 Criterion for Convergence 1.3.1 Monotone Sequence 1.3.2 Application of Monotone Sequence 1.3.3 Cauchy Criterion 1.4 Infinity 1.4.1 Divergence to Infinity 1.4.2 Arithmetic Rule for Infinity 1.4.3 Unbounded Monotone Sequence 1.5 Limit of Function 1.5. 1 Properties of Function Limit 53 1.5.2 Limit of Composition Function 1. 5.3 One Sided Limit 5.4 Limit of Trigonometric Function 1.6 Rigorous Definition of Function Limit 1.6.1 Rigorous Proof of Basic Limits 67 1.6.3 Relation to Sequence Lims of Limit 1.6.2 Rigorous Proof of Propertie 1.6.4 More Properties of Function Limit 1.7 Continuity 1.7.1 Meaning of Continuity 1.7.2 Intermediate Value TheoremContents 1 Limit 7 1.1 Limit of Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Arithmetic Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Sandwich Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.3 Some Basic Limits . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.4 Order Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1.5 Subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2 Rigorous Definition of Sequence Limit . . . . . . . . . . . . . . . . . . 24 1.2.1 Rigorous Definition . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.2 The Art of Estimation . . . . . . . . . . . . . . . . . . . . . . 28 1.2.3 Rigorous Proof of Limits . . . . . . . . . . . . . . . . . . . . . 31 1.2.4 Rigorous Proof of Limit Properties . . . . . . . . . . . . . . . 33 1.3 Criterion for Convergence . . . . . . . . . . . . . . . . . . . . . . . . 37 1.3.1 Monotone Sequence . . . . . . . . . . . . . . . . . . . . . . . . 38 1.3.2 Application of Monotone Sequence . . . . . . . . . . . . . . . 42 1.3.3 Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.4 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.4.1 Divergence to Infinity . . . . . . . . . . . . . . . . . . . . . . . 48 1.4.2 Arithmetic Rule for Infinity . . . . . . . . . . . . . . . . . . . 50 1.4.3 Unbounded Monotone Sequence . . . . . . . . . . . . . . . . . 52 1.5 Limit of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.5.1 Properties of Function Limit . . . . . . . . . . . . . . . . . . . 53 1.5.2 Limit of Composition Function . . . . . . . . . . . . . . . . . 56 1.5.3 One Sided Limit . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.5.4 Limit of Trigonometric Function . . . . . . . . . . . . . . . . . 63 1.6 Rigorous Definition of Function Limit . . . . . . . . . . . . . . . . . . 66 1.6.1 Rigorous Proof of Basic Limits . . . . . . . . . . . . . . . . . 67 1.6.2 Rigorous Proof of Properties of Limit . . . . . . . . . . . . . . 70 1.6.3 Relation to Sequence Limit . . . . . . . . . . . . . . . . . . . 73 1.6.4 More Properties of Function Limit . . . . . . . . . . . . . . . 78 1.7 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 1.7.1 Meaning of Continuity . . . . . . . . . . . . . . . . . . . . . . 81 1.7.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . 82 3