4 ## INFINITY (∞)The definition of "becomes infinite" LET Now, a limit is a number. But infinity, along with its symbol ∞, is not a number, and it is not a place. So to "become infinite," or to say that "the limit is infinity," is simply the language we use to describe a The student should be aware that the word infinite as it is used, and has been used historically, in calculus, does not have the same meaning as in the theory of infinite sets. See the well-known quote by Carl Friedrich Gauss.
DEFINITION 4. "becomes infinite." We say that a variable "becomes infinite" if, beginning with a certain term in a sequence of its values, the absolute value of that term and any subsequent term we might name is greater than any positive number we name, however large.
When the variable is If In both cases, we mean: No matter what large number M we might name, we get to a point in a sequence of values of When the variable is a function Although we write the symbol "lim" for limit, those algebraic statements mean: The limit of Definition 4 is the definition of "becomes infinite;" it is not the definition of a limit. As for the symbol ∞, we employ it in algebraic statements to signify that the definition of becomes infinite has been satisfied. That symbol by itself has no meaning.
Let us see what happens to the values of As the sequence of values of We write: If When a function becomes infinite as Next, let us consider the case when In that case, becomes a very small number, namely 0. We write We should read that as "the limit as See First Principles of Euclid's Elements, Commentary on the Definitions. See especially that a definition is Finally, when In other words, whenever
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At , tan As Limits of rational functions A rational function is a quotient of polynomials (Topic 6 of Precalculus). It will have this form:
where Apart from the constant term, each term of a polynomial will have a factor
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According to 1), above, the limit of each term that contains In similar cases, the first step is:
The result follows on dividing both numerator and denominator by
In other words: Problem 4.
In the following, the rational function is the reciprocal of the one above:
This problem illustrates:
Change of variable Consider this limit: Rather than have the variable approach 0, we sometimes prefer that it become infinite. In that case, we do a change of variable. We put On replacing Where will this come up? In the limit from which we calculate the number e : (Lesson 15.)
Problem 5. In the above limit, change the variable to
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