Fall 2013
MATH 226: Limits and infinite series
Branko Ćurgus
 Wednesday, December 4, 2013

 Thursday, November 1, 2013

 Thursday, October 17, 2013

 In
this pdf file I summarize steps involved in limit proofs for $\displaystyle \lim_{x\to +\infty} f(x) = L$.
 In
this pdf file I summarize steps involved in limit proofs for $\displaystyle \lim_{x\to a} f(x) = L$.
 Tuesday, October 8, 2013


Today we discussed the formal meaning of the phrase "$\displaystyle \lim_{x\to +\infty} f(x)$ does not exist". That is, we formulated the negation of the definition of limit.

A good practice is to review the negations of the definition of a constant function that we gave on Friday, September 27.

Next I will state two definitions of an eventually constant function and the negations of these definitions.

A function $f$ with the domain and the range in ${\mathbb R}$ is an eventually constant function if there exists $c \in {\mathbb R}$ and there exists $X \in {\mathbb R}$ such that for all $x > X$ we have $f(x) = c$.

This definition written using logical symbols: $\exists \, c \in {\mathbb R} \ \ \exists \, X \in {\mathbb R} \ \ \forall \, x > X \ \ f(x) = c$.
 The negation of this definition in English is: A function $f$ with the domain and the range in ${\mathbb R}$ is not an eventually constant function if for every $c \in {\mathbb R}$ and every $X \in {\mathbb R}$ there exists $x > X$ such that $f(x) \neq c$.

The negation written using logical symbols: $\forall \, c \in {\mathbb R} \ \ \forall \, X \in {\mathbb R} \ \ \exists \, x > X \ \ f(x) \neq c$.
 Here is the second definition of an eventually constant function (which is of course equivalent to the first one).
 A function $f$ with the domain and the range in ${\mathbb R}$ is an eventually constant function if there exists $X \in {\mathbb R}$ such that $x > X$ and $y > X$ imply $f(x) = f(y)$.

This definition written using logical symbols: $\exists \, X \in {\mathbb R} \ \ \forall \, x \in {\mathbb R} \ \ \forall \, y \in {\mathbb R} \ \ \bigl( x > X \ \wedge \ y > X \bigr) \ \Rightarrow \ f(x) = f(y)$.

The negation of this definition in English is: A function $f$ with the domain and the range in ${\mathbb R}$ is not an eventually constant function if for every $X \in {\mathbb R}$ there exist $x, y \in {\mathbb R}$ such that $x, y > X$ and $f(x) \neq f(y).$

The negation written using logical symbols: $\forall \, X \in {\mathbb R} \ \ \exists \, x \in {\mathbb R} \ \ \exists \, y \in {\mathbb R} \ \ \ x > X \ \wedge \ y > X \ \wedge \ f(x) \neq f(y)$.
 Friday, September 27, 2013

 Today we discussed the concept of a constant function. We gave two definitions. In the items below $A$ and $B$ are nonempty sets.
 Here is the first definition.

A function $f: A \to B$ is a constant function if there exists $c \in B$ such that $f(x) = c$ for all $x \in A.$

This definition written using logical symbols: $\exists \, c \in B \ \ \forall \, x \in A \ \ f(x) = c$.
 The negation of this definition in English is: A function $f: A \to B$ is not a constant function if for every $c \in B$ there exists $x \in A$ such that $f(x) \neq c$.

The negation written using logical symbols: $\forall \, c \in B \ \ \exists \, x \in A \ \ f(x) \neq c$.
 Here is the second definition. It is interesting to point out that the definition of a
constant function given on Wikipedia is identical to our second definition.
 A function $f: A \to B$ is a constant function if for all $x, y \in A$ we have $f(x) = f(y).$

This definition written using logical symbols: $\forall \, x \in A \ \ \forall \, y \in A \ \ f(x) = f(y)$.

The negation of the Wikipedia definition is: A function $f: A \to B$ is not a constant function if there exist $x$ and $y$ in $A$ such that $f(x)\neq f(y)$.

The negation written using logical symbols: $\exists \, x \in A \ \ \exists \, y \in A \ \ f(x) \neq f(y)$.

One can prove that two definitions are equivalent.

As one more exercise in using logical symbols I will write the definition of the range of a function $f: A \to B$ using logical symbols. The range of $f$ is the following set
\[
\bigl\{ y \in B \, : \ \exists \, x \in A \ \ y = f(x) \bigr\}.
\]
 Thursday, September 26, 2013
