Definitions 10/10/2017

Definition of Domain : Given the function f : A \rightarrow B we say that the domain of f is the set A.

As we discussed today in class the domain is not simply all the values that your function can take in but rather all the values that you decide your function will take in.

Definition \infty is the limit- lim_{x \rightarrow} f(x) = \infty means that for all M > 0 we can find a \delta >0 such that whenever 0< \mid x-c \mid < \delta , it follows that f(x) > M.

An important part of the definition to note is the part that states for all M > 0 we can find a \delta >0 . This line implies that there is a different value of \delta for each value of M which is why in class today we had to find \delta in terms of M in order to complete this proof.

Definition 4.2.1 – Let f : A \rightarrow B, and let c be a limit point of the domain A. We say that lim_{x \rightarrow}f(x) = L provided that, for all \epsilon, there exists a \delta > 0 such that whenever 0< \mid x-c \mid < \delta (and x \in A) it follows that $\mid f(x) -L \mid < \epsilon$.

It is interesting to note the difference between the standard definition for functional limits and the definition for \infty as the limit. This difference is due to the fact that mathematical operations are still not well defined when using \infty. The last part of the standard definition of functional limits uses the expression \mid f(x) -L \mid < \epsilon. We would not want to use this expression where L = \infty if we are attempting to rigidly prove that a limit equals \infty when we cannot even properly define f(x) - \infty .

Theorem 4.2.3- Given f: A \rightarrow \mathbb{R} and limit point c \in A, the following statements are equivalent.

i) lim_{x \rightarrow c} f(x) = L

ii) For all sequences (x_n) \subseteq A such that x_n \neq c, it follows that f(x_n) \rightarrow L.

Since i and ii are equivalent this theorem is used to to define the divergence criterion for functional limits. Mainly the divergence criterion utilizes ii. It says that if two sequences x_n and y_n exist for a function f such that x_n \neq c (and c is a limit point), y_n \neq c and lim x_n = lim y_n = c but if lim f(x_n) \neq lim f(y_n) we can say lim_{x \rightarrow} f(x) does not exist. Here we see the usefulness of Theorem 4.2.3 because it allows us to split lim_{x \rightarrow c} f(x) = L into two requirements that we can then use to define divergence for functional limits.

Posted on by Abraham Schroeder
Categories: Definitions

One Response

  1. Jeremy LeCrone says:
    October 17, 2017 at 4:44 pm

    Thank you for the exploration of concepts covered in class, Abe. Good blog post.

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