Limits of Functions - Single-Variable Calculus (Undergraduate Foundation)

This course provides a rigorous introduction to the concept of the limit, the theoretical bedrock upon which all of calculus is built. We will move from an intuitive understanding of what it means for a function to 'approach' a value to the algebraic techniques required for precise evaluation. The course covers the limit laws, methods for handling indeterminate forms, and the behavior of functions at infinity. Understanding limits is non-negotiable for any serious study of calculus. The principles developed here are not merely abstract; they are the tools used to formally define the core concepts of calculus that are used to model the mechanics of change that govern engineering, physics, economics, and computer science. By the end of this course, you will be able to: evaluate limits graphically, numerically, and algebraically; apply the Squeeze Theorem and L’Hôpital’s Rule; analyse the end-behavior of functions; and identify vertical and horizontal asymptotes. This course is designed for first-year undergraduates in STEM fields who are beginning their calculus sequence. It is an essential prerequisite for subsequent courses on continuity and differentiability and is also invaluable for any student or professional seeking to rebuild their mathematical foundation from first principles.

146

19 hrs

Payment required for enrolment
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.
[FUTA, Akure] MTS 102: Introductory Mathematics II
[FUTA, Akure] MTS 102: Introductory Mathematics II
This learning track is structured for first-year students at the Federal University of technology, Akure (FUTA) and mirrors the standard second-semester coverage of elementary calculus. It begins with single-variable functions and their graphs, then walks learners through limits, continuity, differentiation techniques, and curve sketching—just as covered in the official MTS 102 outline. You’ll also explore anti-derivatives and integration, learning both the techniques and how to apply them to solve practical problems in science and engineering contexts. Everything is broken down into short, focused video lessons that keep things clear and manageable, especially for students who might be engaging this content for the first time. If you're not a FUTA student but need to build a solid foundation in these same topics, this track can serve you just as well. The structure and explanations are universal, ensuring that learners with similar academic goals can benefit fully.

This learning track is structured for first-year students at the Federal University of technology, Akure (FUTA) and mirrors the standard second-semester coverage of elementary calculus. It begins with single-variable functions and their graphs, then walks learners through limits, continuity, differentiation techniques, and curve sketching—just as covered in the official MTS 102 outline. You’ll also explore anti-derivatives and integration, learning both the techniques and how to apply them to solve practical problems in science and engineering contexts. Everything is broken down into short, focused video lessons that keep things clear and manageable, especially for students who might be engaging this content for the first time. If you're not a FUTA student but need to build a solid foundation in these same topics, this track can serve you just as well. The structure and explanations are universal, ensuring that learners with similar academic goals can benefit fully.

[UNILAG, Akoka] MTH 102: Elementary Mathematics II
[UNILAG, Akoka] MTH 102: Elementary Mathematics II
This learning track is designed to guide first-year students at the University Of Lagos through key concepts in calculus, beginning with the fundamentals of single-variable functions and their graphs. It builds gradually into the core topics of limits, continuity, and differentiability, with each course tailored to simplify these foundational ideas for early learners. The focus is not just on theory but also on building the skill to solve problems confidently, especially those typically encountered in university-level exams. You’ll move from understanding the concept of a limit to mastering how derivatives work and how to apply them to sketch curves and analyze function behavior. Although built for UNILAG students, this track is suitable for anyone looking to strengthen their understanding of introductory calculus at the university level. Whether you're preparing for school assessments or seeking a solid refresher, this track will help you follow a structured path.

This learning track is designed to guide first-year students at the University Of Lagos through key concepts in calculus, beginning with the fundamentals of single-variable functions and their graphs. It builds gradually into the core topics of limits, continuity, and differentiability, with each course tailored to simplify these foundational ideas for early learners. The focus is not just on theory but also on building the skill to solve problems confidently, especially those typically encountered in university-level exams. You’ll move from understanding the concept of a limit to mastering how derivatives work and how to apply them to sketch curves and analyze function behavior. Although built for UNILAG students, this track is suitable for anyone looking to strengthen their understanding of introductory calculus at the university level. Whether you're preparing for school assessments or seeking a solid refresher, this track will help you follow a structured path.

[OAU, Ife] MTH 201: Mathematical Methods I
[OAU, Ife] MTH 201: Mathematical Methods I
Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

1. Introduction
2

Meaning of the limits of a real-valued functions - graphical illustrations and formal (epsilon-delta) definitions.

Chapter lessons

1-1. Informal definition
19:20

An informal definition of the limit of real-valued functions.

1-2. Formal definition
17:26

A formal (rigorous) definition of the limit of real-valued functions.

2. Proving Finite Limits
5

How to prove that a given finite limit as x approaches a finite value is correct, using the formal or epsilon-delta definition.

Chapter lessons

3. Evaluating Finite Limits (1)
5

Different methods of evaluating limits of real-valued functions - direct substitution, theorems, graphing, factorization, conjugates.

Chapter lessons

3-1. Direct substitution
23:58

Evaluating limits of functions by direct substitution.

3-2. Theorems
32:03

Evaluating limits of functions by use of fundamental theorems.

3-3. Graphing / use of calculator
35:01

Evaluating limits of functions by graphing or use of a calculator.

3-4. Factorization
19:30

Evaluating limits of functions by factorization.

3-5. Conjugates
13:00

Evaluating limits of functions by use of conjugates.

4. Evaluating Finite Limits (2)
4
2

Other methods of evaluating finite limits - L’Hôpital’s rule, use of known special limits, squeeze theorem, limits of piecewise-defined functions, etc.

Chapter lessons

4-1. L’Hôpital’s rule
45:15

Evaluating limits of functions by L’Hôpital’s rule.

4-2. Special limits
27:58

Evaluating limits of functions by use of known special limits.

4-3. The squeeze theorem
34:06

Evaluating limits using the squeeze theorem.

4-4. Piecewise-defined functions
22:20

Evaluating limits of piecewise-defined functions.

5. Evaluating Finite Limits (3)
1
2

Intuitive methods of evaluating finite limits of real-valued functions.

Chapter lessons

5-1. What if everything fails?
45:39

Examining limits for which the L'Hospital's rule fails.

6. Proving Limits Involving Infinity
2
2

Formal and informal definitions and proofs of limits at infinity and infinite limits.

Chapter lessons

6-1. Informal definition
42:51

Informal definition of limits at infinity and infinite limits.

6-2. Formal definition
15:13

Formal (rigorous) definition of limits at infinity and infinite limits.

7. Evaluating Limits at Infinity
8
3

Different methods of evaluating limits of real-valued functions at infinity and infinite limits.

Chapter lessons

7-1. Polynomials and rational functions
30:49

Evaluation of limits at infinity for polynomials and rational functions.

7-2. Exponential functions
30:11

Evaluation of limits of exponential functions at infinity.

7-3. Special limits
20:30

Evaluation of limits at infinity using a special limit.

7-4. Rationalization
21:03

Evaluation of the limits of functions with rationalizable radicals at infinity.

7-5. Modulus function
30:19

Evaluation of the limits of functions with non-rationalizable radicals at infinity.

7-6. Indeterminate forms (1)
12:41

Indeterminate form ∞-∞.

7-7. Indeterminate forms (2)
38:19

Indeterminate form 0*∞.

7-8. Indeterminate forms (3)
33:42

Indeterminate forms involving powers.