Math 199 (Assignments)

[To be turned in on Monday]

Consider the definition of the limit of a function at a point:

Definition: Suppose $f$ is a function and $c$ and $L$ are numbers. Then $\displaystyle\lim\limits_{x\to c}f(x) = L$ means that, for every $\epsilon > 0$, there is a $\delta > 0$ such that $|f(x)-L| < \epsilon$ whenever $0<|x-c|<\delta$.

1. Translate the definition of the limit of a function below into formal notation. (Translate only the part after "means that".)
   
   Note: This is not a word-by-word translation, but a translation of concepts. It will turn out to be a heavily nested set of propositions, so it is not a simple expression, and you will have to find a way to deal with the issues of being greater than 0, for example. Be sure that all of the letters that you use have a meaning in your formal statement; they must either be given ahead of time (like $f$), or introduced by a quantifier.
   
   This one is not easy, to you may have to take some time thinking about it.
2. Write the negation of this statement in formal notation.
3. Translate the negation into words. The best answer will be one that is not just a symbol-by-symbol transliteration, but one that more naturally expresses the meaning of the expression.

Math 199 (Fall 2019) web pages Created: 26 Sep 2019 Last modified: 26 Sep 2019 at 8:52 PM Comments to: dpvc@union.edu

Assignments