Subsidiary Mathematics

My goals

 By the end of this unit, I will explain: 

. Concepts of limits. 

. Indeterminate cases. 

. Applications.

Introduction

The limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular point.

Suppose that a person is walking over a landscape represented by the graph of the function f (x). His/her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. His/ her altitude is given by the coordinate y. He/she is walking towards the horizontal position given by x = p. As he/ she gets closer and closer to it, he/she notices that his/her altitude approaches L. If asked about the altitude of x = p, he/she would then answer L. It means that her/his altitude gets nearer and nearer to L except for a possible small error in accuracy.

The limit of a function f (x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f (x) remain within the target distance.

Limits are also used to find the velocity and acceleration of a moving particle.

Required outcomes 

After completing this unit, the learners should be able to: 

» Calculate limits of certain elementary functions. 

» Apply informal methods to explore the concept of a limit including one sided limits.  

» Solve problems involving continuity. 

» Use the concepts of limits to determine the asymptotes to the rational and polynomial functions. 

» Develop calculus reasoning.

1. Concepts of limits 

Neighbourhood of a real number 

Activity 1

Study the following political map of Lesotho, Swaziland and South Africa. What can you say about the boundaries of Lesotho and Swaziland?

Exercise 1

1. Apart from The Kingdom of Lesotho, give two examples of countries or Cities in the world that are surrounded by a single country or city. 

2. Give three examples of intervals that are neighbourhoods of -5? 

3. Is a circle a neighborhood of each of its points? Why? 

4. Draw any plane and show three points on that plane for which the plane is their neighborhood.

Note:

A deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function. 

Limit of a function 

Activity 2

d) Division:

A really large (positive or negative) number divided by any number that is not too large, is still a really large number and we have to be careful with signs.

Beware!

So, we have dealt with almost every basic algebraic operation involving infinity. There are three cases that we have not dealt with yet. These are 

To find a limit graphically, we must understand each component of the limit to ensure the graph is used properly to evaluate the limit.