Defining a Limit

Defining a Limit In truth, there is a large distance between the most intuitive definition of a limit, and the formal definition of the limit; the intuitive definition is deficient, and inaccurate in a number of ways. By considering these deficiencies, we can try to find better definitions for a limit, hopefully motivating the use of the formal definition. In the following discussion, we will move from the most coarse, to most refined definition of the limit. 1. When we write limx→x0 f (x) = L we mean that as the input values x approach the point x0 , the output values of the function f (x) approach L. This is in a sense the most crude understanding of what a limit is. It is based on a perception that for a reasonable function, it seems like there is a value that the function ought to have at the point x0 , which is the limit L - whether or not the function is actually defined so that f (x0 ) = L. However, there are some major problems with this definition. First of all, this definition does not indicate the fact that the value of the function at the point x0 , f (x0 ), has no bearing on whether or not the limit exists, or what it exists as. Second, the terms approach and approaches are much too vague for us to really work with them with any amount of accuracy. It is reasonable to assume that when we say approach, we mean ‘get closer to.’ Thus, this definition says as the input values x get closer to x0 , the function values f (x) get closer to the limit value L. However, there is no mention of how close the function values need to get to the limit! Consider the example f (x) = −x2 which is a parabola opening downward, with its vertex at the origin of the Cartesian coordinate plane. We know that lim (−x2 ) = 0 x→0 yet based on our current definition, we could just as well conclude that lim (−x2 ) = 5 x→0 or any other positive number. As x → 0, clearly f (x) gets closer to 5, which means that the limit is 5, according to the above definition. In this way we have an infinite number of different values for the limit, yet the limit itself must be unique! This is a serious problem with our definition, because there is nothing about the value 5 (or any other positive number) that makes it seem it ought to be the value of f (x0 . Since this defintion falsely classifies values as limits which are not, and does not require the limit to be unique, it is not precise enough to be of any real use in mathematics. It is a reasonable intuitive notion to keep in mind, but it is not a sufficient foundation to build mathematics upon. 2. When we write limx→x0 f (x) = L we mean that we can force the output values f (x) to be arbitrarily close to, but not necessarily equal to, the limit value L, by choosing x arbitrarily close to, but not equal to x0 . Before we can proceed we need to define the term arbitrary close, and what exactly it means in terms of the definition. When we say arbitrarily close, but not equal to, we mean for any positive distance, we can make the values in question closer. That is, we can make the quantities as close as anyone could possibly desire, although not necessarily equal. In terms of f (x) values approaching L, it means that we can approximate L with the values f (x) to any degree of accuracy; we can make the distance between f (x) and L less than any positive error tolerance of approximation. In terms of the x values approaching x0 , the term arbitrarily close has a slightly different meaning. With our notion of a limit, only the points x very close to x0 should influence what the limit value is. In this case, very close means as close as anyone could possibly desire; given a positive distance, we can find x values closer to x0 than that distance, so that the relationship between f (x) and L is satisfied. Note that we do not allow x = x0 , but we simply do not require that f (x) = L, but it is acceptable if it does. Also note that sometimes the term sufficiently close may be used interchangeably with arbitrarily close; ie. the two terms have the same meaning. In light of the previous discussion, this is a much better definition. First, it illustrates the fact that the value of f (x0 ) has no bearing on the existence or value of the limit, because we don’t care about the point x = x0 . Second, it clears up the problem with the previous example, insofar that lim (−x2 ) = 0 x→0 by this definition, as it ought to. Nevertheless, this definition still does not guarantee uniqueness of a limit. For instance, consider the function f (x) = |x| x which is 1 for x > 0 and -1 for x < 0. Using this definition we can conclude both lim f (x) = 1 and x→0 lim f (x) = 1 x→0 ince we can choose x values arbitrarily close but not equal to 0, so that f (x) is both arbitrarily close (in fact equal to) 1 or -1. Although this definition is an improvement, it is still not sufficient, as it does not guarantee uniqueness of a limit. 3. When we write limx→x0 f (x) = L we mean that we can force the output values f (x) to be arbitrarily close to, but not necessarily equal to, the unique limit value L, by choosing x arbitrarily close to, but not equal to x0 . This is exactly the same definition as above, with a single addition - the word unique. Now in this definition we require that the limit value L be uniquely related to the function f and the point x0 in the way described by the definition. This is more promising, as we can conclude that limx→0 |x|/x does not exist, as there is no unique value that satisfies the relationship (there are two). It even gives us the proper result for a slightly more contrived example, such as the function  1 x = 1/n for somen ∈ N f (x) = 0 otherwise Here is a function that is 0 almost everywhere, except for these points where the input value x = 1/n, where n is a natural number. Thus, we have a function that is 0, but that jumps up to 1 every now and then. The thing to notice is that if we look at the limit as x → 0, we can always find x values as close as we want to 0 so that f (x) = 0 or f (x) = 1. Thus, the limit as x → 0 should not exist, which this definition properly classifies. In fact, any limit that satisfies this definition, will satisfy the formal definition of the limit. However, there is still one deficiency in this definition. It gives us no means of proving mathematically that a limit exists. While we can make this type of verbal argument for a simple example, it is simply insufficient for more complicated examples. Thus, we move from this definition to the formal definition not because this definition is inaccurate, but because the formal definition provides us with a stronger tool for mathematical proofs. In order to further motivate the formal definition of the limit, we will introduce yet another equivalent definition for the limit. 4. When we write limx→x0 f (x) = L we mean that L is the unique number which can be approximated as accurately as desired by evaluating the function f (x) using x-values close enough to, but not equal to x0 . In this definition we once again require uniqueness. Here we replace the term arbitrarily close with approximated as accurately as desired, but the two phrases have essentially the same meaning. For any distance away from L, we can force the function values to be within that distance; ie. approximate the limit value L with any degree of accuracy. This wording may be preferable because it emphasizes that at the heart of limit concept is approximation. When we use the words close enough to, we have the exact same meaning as sufficiently close or arbitrarily close to. We will not consider any examples with this definition, as it behaves identically to the previous one. 5. We say that limx→x0 f (x) = L if for any error tolerance around L, we can find an interval around x0 (excluding x0 ) such that for all x values within the interval, f (x) approximates L within the error tolerance (ie. the distance between f (x) and L is within the error tolerance). This is a verbal statement of the formal definition of the limit, which is still a deficient tool for mathematical proof. Noteworthy is that we no longer needed to explicitly state uniqueness, as it is guaranteed by this definition. By recasting this definition into variables, we have an extremely powerful, yet somewhat more difficult tool to uderstand. Definition: The Limit Suppose f (x) is defined on an open interval about x0 , not necessarily containing x0 . We say that L is the limit of f (x) as x approaches x0 , written lim f (x) = L x→x0 if for every number  > 0, there exists a corresponding number δ > 0 such that for all x with 0 < |x − x0 | < δ we have |f (x) − L| < . This definition is the same as the definition we stated above, but we have rewritten some of the verbal terms such as ‘error tolerance’ in terms of variables. We can describe the mathematical definition in words as follows: for any error tolerance ( > 0), we can restrict the values of f (x) to be within that error tolerance of L (|f (x) − L| < ) by restricting the values of x to be sufficiently close but not equal to x0 (by choosing δ > 0 and considering x with 0 < |x − x0 | < δ). We require 0 < |x − x0 | in order to exclude the point x0 from the interval of interest, because the limit does not depend on the function value at that single point. It is also noteworthy that when we use the phrase ‘for every  > 0,’ it is sufficient to show that it is true for an arbitrary  (because we could just as well replace ‘this’  with any and ‘every’ number  and the analysis would still hold true). We have recasted the verbal definition so that we can manipulate the variables  and δ in order to make precise calculations to prove whether or not a limit exists. Let us consider the following examples in order to further understand this definition.