Functions, Limits and Continuity - Calculus (Undergraduate Foundation)

Calculus starts with understanding how one thing changes in relation to another. This course defines functions, explores domains and ranges, and teaches you to plot graphs accurately. You will master limits, including one-sided limits, to understand how functions behave as they approach specific points. The study concludes with the strict conditions required for a function to be continuous. These concepts are the bedrock of all advanced mathematics and engineering. Engineers use these tools to predict how structures hold weight or how machines move. Scientists apply limits to study chemical reactions and biological growth rates over time. In economics, these functions help calculate profit margins and market trends. Mastering this foundation allows you to solve real-world problems involving motion, fluid flow, and financial forecasting. It turns abstract numbers into practical data for building and managing systems. You will learn to identify different function types and determine their valid input and output values. The course provides skills to sketch various graphs and calculate limits using algebraic methods. You will gain the ability to check if a function is continuous at any given point or over an interval. By the end, you will be able to handle complex calculus problems with speed and precision. This knowledge prepares you for more advanced topics like derivatives and integrals. This course is for university freshers in engineering, science, or economics who need a solid math foundation. It also suits secondary school leavers preparing for university entrance exams or technical diplomas. Even those in non-science fields will benefit by developing the logical thinking and analytical skills needed for data-driven roles. Anyone looking to refresh their basic calculus knowledge for professional exams will find the direct approach useful. The material is structured to move you from basic definitions to technical mastery quickly.

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

AMS 102: Basic Mathematics

Management and administration require precise numerical logic for decision-making. This track follows the official NUC CCMAS syllabus for AMS 102, covering real numbers, set theory, and complex systems. You will master algebraic operations, permutations, and combinations alongside the foundations of trigonometry and calculus. It provides the mathematical tools to model financial trends, handle business data, and solve optimisation problems in corporate settings. This programme is built for first-year university students in Accounting, Business Administration, Banking, Finance, and Public Administration. It also serves secondary school leavers preparing for management degrees and professionals needing a refresher on business mathematics. It is suitable for anyone moving into roles that demand accurate quantitative analysis and logical deduction. You will gain the ability to simplify complex business formulas, calculate probabilities for risk assessment, and find the exact points where profits are highest. You will be able to handle sequences for interest calculations and use differentiation to manage production costs. Completing this track ensures success in university examinations and builds the analytical strength required for careers in financial analysis, auditing, and strategic management.

MTH 102: Elementary Mathematics II - Calculus

Calculus is the mathematical tool for measuring change and finding the best results in any system. This track follows the official NUC CCMAS MTH 102 curriculum to teach you how functions behave, how to calculate exact rates of change, and how to sum up tiny movements to find total areas or volumes. You will move from basic limits to complex integration techniques used to solve practical problems in the physical world. This track is for first-year university students in engineering, science, and economics departments across Nigeria. It also serves secondary school leavers preparing for technical degrees or university entrance exams. Anyone needing to build a strong foundation in mathematical logic for data analysis or professional licensing will find these lessons essential. After finishing this programme, you will calculate derivatives using first principles and shortcuts, find turning points for optimisation, and solve definite integrals for land measurement or tank design. You will use the trapezium and Simpson's rules to handle experimental data accurately. These skills ensure you pass your university exams and excel in technical careers like civil engineering, physics research, or financial forecasting.

Course Chapters

1. Introduction

This chapter defines functions and rules for mapping inputs to outputs. Accurate modelling requires mastering these basics to prevent errors in advanced calculus. These concepts form the essential bedrock for limits and derivatives. You will learn to define functions; determine domain and range; and use interval notation. You will also evaluate piecewise functions and simplify the difference quotient.

Concept Overviews

6 Lessons

2:09:37

Problem Walkthroughs

6 Lessons

36:21

2. Graphs

Graphs convert equations into visual tools for engineering and science. This chapter teaches you to plot functions accurately on coordinate axes to predict trends and solve problems. Master sketching polynomial, trigonometric, and rational graphs; apply shifts and reflections; identify intercepts and asymptotes; and draw piecewise functions with breaks.

Concept Overviews

7 Lessons

4:29:33

Problem Walkthroughs

5 Lessons

56:39

3. Operations

Functions can be combined and reversed to create complex mathematical relationships. This chapter teaches you how to manipulate these models, which is a vital skill for solving multi-step engineering and science problems. You will master calculating arithmetic combinations of functions; solving and decomposing composite equations; deriving inverses for linear and rational functions; and applying the reverse order rule to invert multi-step operations.

Concept Overviews

5 Lessons

1:48:53

Problem Walkthroughs

6 Lessons

33:20

4. Introduction to Limits

Limits explain function behaviour near specific points, especially where they are undefined. This concept is the mandatory starting point for derivatives and integrals. You will learn to predict output values by observing trends from both sides of an input. You will master calculating limits using algebraic laws; evaluating one-sided and infinite limits; verifying limit existence; and identifying horizontal and vertical asymptotes to determine end behaviour.

Concept Overviews

6 Lessons

2:21:51

Problem Walkthroughs

6 Lessons

51:08

5. Evaluation of Limits (1)

This chapter teaches precise limit evaluation. Direct substitution works first. Algebraic methods then clear undefined forms. These skills build the foundation for all future calculus topics in this course. You will learn to use direct substitution, factorise expressions to clear zero divisions, rationalise radicals to remove roots, and compare polynomial degrees for infinite limits.

Concept Overviews

4 Lessons

Problem Walkthroughs

12 Lessons

6. Evaluation of Limits (2)

This chapter handles limit problems that break standard substitution. Mastering these edge cases prevents calculation dead ends and builds reliable analytical skills. You will apply the squeeze theorem to bounded functions, calculate special trigonometric limits, transform shifted angle identities, and verify limit existence at dense piecewise junctions.

Concept Overviews

3 Lessons

Problem Walkthroughs

9 Lessons

7. Continuity

A function is continuous if its graph can be drawn without lifting your pen. This chapter defines the mathematical conditions required for continuity and introduces existence theorems like the Intermediate Value Theorem. Identifying discontinuities is critical for ensuring the stability of engineering and physical systems. You will learn the three conditions for continuity; identify different types of discontinuity; solve for constants that make a function continuous; and use theorems to prove the existence of roots.

Concept Overviews

4 Lessons

Problem Walkthroughs

4 Lessons

8. Conclusion

This final chapter consolidates everything you have learned about functions, limits, and continuity. It serves as the final bridge to the next stage of your calculus journey. Reviewing these foundations ensures you are ready for differentiation and integration. You will summarise the core principles of functions; review limit evaluation techniques; and confirm the conditions for mathematical continuity across different function types.

Concept Overviews

1 Lesson