Chapter 5: Limits and Continuity

Limits of Functions

The limit concept is at the heart of differential and integral calculus and it is the aim of this chapter to introduce the limit concept and define the limits of functions. Limit is the number which the value of a function “approaches” as its argument (variable) approaches to a certain point. In this context “approach” means that as the variable gets arbitrarily close to a particular point, but never reaches it, the function 31 may get sufficiently close to a certain value, say 31. Since, what is important is the behaviour of the function 31 around this particular point, say31, the value of the function at31 is irrelevant to the limit concept. Therefore, the definition of the limit of the function 31 at a given point 31 is that 31 can be chosen arbitrarily close to a number 31 by taking 31 sufficiently close to 31but not equal to 31. We then say that the limit of 31 as 31 approaches 31 is 31 and we write it as

31

where 31 indicates that 31 approaches the point31. We read 31 as ''The limit of 31 as 31 approaches 31 is 31''. Thus, if there is such value L, 31 may not be defined at the point31, we can make the value of 31 as close as we want to 31. In other words, the limit of a function at a given point 31determines the behaviour of the function near 31.

To understand this concept, we want to determine the behaviour of 31 around the point 31. That is, what does the value 31 approach when 31 tends to the point 31? Now, let us consider points 31 around 31 such that the distance between points 31 and 31 gets smaller as seen in the Table below.

31 In this case, from the table given above, as the numbers 31 get closer and closer to the point 31 (either with smaller or greater values than 31), since the value 31 appear to be sufficiently close to the value 4, we estimate that limit is 31and we say that 31 is the limit of the function 31 as 31 approaches the point 31. In mathematical notation, this statement is written in the form

31or

31

If 31, the following figures explain there will only be three different cases. In the first one, 31, i.e. the function is defined at 31 and has the same value as its limit; in the second case, 31is not defined but it has a limit value; in the last one, the function is defined but the value of 31 is different than its limit value, i.e. 31.

31 31 31

One –Sided Limits

Although a function may have a limit at a given point, it is sometimes enough to know the behaviour of the function to the right or left of a point 31. In this step, we just need to consider the approach of 31 to the point 31 from only one side.

If the approach is from one-side, say from the right of 31, we use the notation 31; if it is from the left of 31, we use the notation 31. In the case, we say that the right limit of 31 is 31 and it is denoted by

31,

and that the left limit of 31 is 31 and this is denoted by

31

respectively. Note that there is important relationship between two-sided and one-sided limits and this relationship can be given as follows:

31if only if 31 and 31.

Now, we consider two significant examples which we will frequently be used to calculate the limits of various functions:

Consider the constant function31, where31. For any 31,

31

Let 31 be the unit function and 31. For this function, we have

31

Rules for Calculating Limits

Now, we give the following properties of limits which make it easy to calculate limits of the functions encountered in this chapter provided that the limits of elementary functions are known.

Limit rules:

Suppose that 31 is a constant and the limits 31, 31exist and are 31 and 31, respectively. Then

1. limit of a sum: 31, that is the limit of a sum is the sum of the limits.
2. limit of a difference:31, namely the limit of a difference is the difference of the limits.
3. limit of a product: 31, that is the limit of a product is the product of the limits.
4. limit of a constant multiple:31, that is the limit of a constant multiple is the product of the limit by the constant.
5. limit of a quotient: 31, 31that is the limit of a quotient is the quotient of the limits provided that the limit of the denominator is not 0.

Note that limit rules are applicable to one sided limits by the symbolism 31 can be replaced by 31or 31.

Infinite Limits and Limits at Infinity

Now, we are going to extend the idea of limit. We will consider limits involving the concept of infinity. We employ the symbols 31and 31 to indicate that either a variable or a quantity unboundedly grows in the negative and positive direction, respectively. We will consider two kinds of limits:

1. infinite limits
2. limits at infinity

Let 31 be a function defined about the point 31, except possibly at 31itself. If 31 takes arbitrarily large values (31 increases without bound) as 31 approaches 31then we say that the limit of 31 as 31 approaches 31, is infinity. It is denoted symbolically by

31

In a very similar manner, we have the following: if, as 31 approaches 31, 31 takes arbitrarily large negative values, i.e. 31 increases without bound in the negative direction,31then we say that the limit of 31 as 31 approaches 31, is minus infinity and denote it symbolically by

31

Similarly, by making use of the limit introduced here, we may answer the interesting question:

“How does a function 31 behave as its variable 31 become arbitrarily large, i.e. what happens to 31 as 31 or 31” ?

Symbolically we write 31and say that the limit of 31 is 31, if it exists, as 31 approaches infinity or negative infinity. Such a number 31 may not exist which indicates that 31 approaches 31 or 31.

Recall that the symbols 31and 31do not represent numbers. Actually, they imply unbounded behaviour when used in the limit concept.

Note that if the function is a polynomial the term with the highest degree of the polynomial determines its behaviour. That is, the term with the highest degree dominates the remaining terms as 31 becomes large. Also, the limit which is expressed as31 (31provided that 31 makes sense ) is always true.

Continuity

Finally, the continuity of a function which is a far-reaching and important concept in calculus is shortly discussed. Using the limit, we define the continuity of a function 31 at a given point.

A function 31 is called continuous at the point 31if

31

Note that if 31 is continuous at the point 31, the following three conditions must be ensured:

a.) 31is defined, that is, 31is in domain of 31,

b.) 31exists,

c.) 31

Thus, the graph of a continuous function at a given point 31of its domain has no breaks, holes, or jumps at this point; in other words, the graph of a continuous function through the point 31can be drawn without lifting the pencil from the paper.

Note that, despite a function 31may not have a limit at 31in its domain, it can still have one-sided limits at 31 (that is, the function 31 may have a jump at 31). Thus, in this case, we can extend the definition of continuity:

If 31, we say that 31 is right continuous at 31and if 31, we say that 31 is left continuous at 31.

Now we can give a relation between continuity and one-sided continuity of a function 31 in the following manner:

A function 31 is continuous at 31if only if it is right and left continuous at this point.

If 31 is not continuous at 31or if 31is not defined, we say that 31 is discontinuous at 31. That is, if any one of the three conditions listed above are not satisfied, then31 is said to be discontinuous at 31.

In the case of complicated functions (sums, quotients, composite, etc.) we can make use of the properties of limits (see, Limit Rules) to give significant properties that may help us determine the continuity of given functions. That is if the functions 31and 31 are continuous at a point 31, then the sum 31, the difference 31, the product 31 and the quotient 31(provided that 31) are continuous at 31.

Significant properties of continuous functions mentioned in this chapter is that if a function 31 is continuous on the closed, bounded interval 31 must have an absolute maximum value and an absolute minimum value on the interval 31.