Due: November 17th, 2017
Math 142A Assignment 6
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For each of the following, evaluate the limit indicated or prove that it does not exist.
 $\displaystyle\lim_{x \to 3} \frac{\sqrt{3x}3}{x3}$
 $\displaystyle\lim_{x \to \infty} \frac{3x^4+6x^312x+3}{(2x+1)(5x^2+4x+16)(x2)}$
 $\displaystyle\lim_{x \to 0} \frac{3x^4  2x^3 + x^2 + 10x}{(4x^2x)(x^5+x^3+x+2)}$
 $\displaystyle\lim_{x\to0^+}\displaystyle\lim_{y \to 0} x^y$
 $\displaystyle\lim_{y\to0}\displaystyle\lim_{x \to 0^+} x^y$

Suppose $f, g : S \to \mathbb{R}$ and $h : f(S) \to \mathbb{R}$ are monotonic functions. For each of the following functions, either prove it is monotonic or provide an example to show that it is not necessarily monotonic.
 $f+g$
 $fg$
 $h\circ f$
 $\frac1{f}$, assuming $0 \notin \color{red}{f(S)}$


Suppose that $S \subset \mathbb{R}$ is bounded above, $f : S \to \mathbb{R}$ is monotonic, and $x = \sup S$.
Show that the limit from the left of $f$ at $x$ either converges, diverges to $\infty$, or diverges to $\infty$.

As a consequence of the above, show that if $S \subset \mathbb{R}$ is arbitrary (i.e., potentially unbounded) and $ x \in \mathbb{R}$ is the limit of some increasing sequence in $S$, then the limit from the left of $f$ at $x$ either converges, diverges to $\infty$, or diverges to $\infty$.

Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous. Show that $f$ is monotonic if and only if $f^{1}(\{y\})$ is an interval (although potentially empty) for every $y \in \mathbb{R}$.

Evaluate the following limits:
 $\displaystyle\lim_{x\to1}\frac{x^31}{x1}$
 $\displaystyle\lim_{x\to1}\frac{x^n1}{x1}$, for $n \in \mathbb{N}$
 $\displaystyle\lim_{x\to1}\frac{x^nx^m}{x1}$, for $n, m \in \mathbb{N}$

Bonus:
A set $S$ is said to be countable if there is some map $f : S \to \mathbb{N}$ which is onetoone.
In the first assignment, you showed that $\mathbb{Q}_+$ and $\mathbb{N}^2$ were countable.
Suppose $f : \mathbb{R} \to \mathbb{R}$ is monotonically increasing. Show that the set of points at which $f$ is discontinuous is countable.