This book is designed as an advanced guide to Differential Calculus. This covers the following topics: introduction and strict definition of limits, properties of limits, limits of trigonometric functions, infinite limits and limits to infinity, continuity of functions, miscellaneous of exercises. This is including the deduction of many formulas to clarify concepts and fill all the needs of the students to achieve the fundamentals of Limits and Continuity of Functions. Finally, another objective of the book is to achieve the interest and motivation of the students to the real nature of Limits and Continuity of Functions.