In Part I of this series of articles, we introduced the concept of limit and discussed how this seemingly simple concept launched a branch of mathematics which would come to be known as the calculus. As we learned, the limit is nothing more than a value which is approached by a function when we let the independent variable become arbitrarily close to some value. Aside from defining the derivative-an essential function in all of calculus-the limit allows us to talk about things like division by zero. How strange!
In the first article, we looked at the function y = 1/x and examined what the limit of this function was when x approached 2 and even more interestingly when x approached 0. The latter investigation compelled us to realize what division by zero actually meant: that of infinity. In other words, division by zero is classified as undefined because we cannot put a set value on infinity. Infinity by its very nature is undefined. After all, how big is infinity? You can think of the vastness of space, and its size still pales in comparison to ultimate infinity. Yes, indeed. Infinity is a very strange concept. (See more on this in my series of articles Dabbling in Infinity)
After reading the first article, you still may be wondering what is so unobvious about this concept of limit, for in the example of y = 1/x, with the exception of x approaching 0, any other limit value can be computed by direct substitution. Thus the limit of this function when x approaches 3 is 1/3. The limit when x approaches 10 is 1/10, and so on. However, if we examine the rational function
y = (x^2 + 5x + 4)/(x + 4) and ask what is the limit of this function when x approaches -4, we cannot compute this by direct substitution as the value of the function is undefined here since the denominator becomes zero. What we do in this case is use an algebraic simplification of the rational function and write
(x^2 + 5x + 4)/(x + 4) as (x + 4)(x + 1)/(x + 4). We then use the cancellation property of division to obtain
(x + 4)(x + 1)/(x + 4) = x + 1 (Canceling the (x+4) from both the numerator and denominator). Now we can examine what happens to this algebraically equivalent function at values distinct from -4. For when x is not equal to -4 but gets sufficiently close to -4, then the limit of the original rational function will approach x + 1 or -4 + 1 or
-3.
Remember. The limit concept does not mean that the independent variable gets to equal the value in question but only approach it sufficiently close. This idea, in effect, seems to give us something for nothing. We cannot divide by zero, for example, but we can divide by numbers that get so close to zero, that in effect, they are as close to zero as almost being zero themselves. In the example of the rational function just discussed, we cannot talk about the value of the function when x = -4 but we can talk about its value when x takes on values sufficiently close to -4.
This "something for nothing" largesse, granted to us by the concept of limit, opens up a whole realm of higher mathematics and gives us the ability to calculate quantities that, without such a grand idea, would be simply incalculable. Thus through the limit we move to the derivative, a special limit we will discuss in an upcoming article, and from here to the antiderivative or integral, from which we can do such things as calculate irregular areas and even volumes. Quite a grant from such a simple concept!
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Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC's of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity; the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.
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