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.

147

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.
[OAU, Ife] MTH 201: Mathematical Methods I
[OAU, Ife] MTH 201: Mathematical Methods I
This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

[NUC Core] GET 209: Engineering Mathematics I
[NUC Core] GET 209: Engineering Mathematics I
Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses. This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory. Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses. This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory. Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

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.