Due: November 17th, 2017

Math 142A Assignment 6

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. For each of the following, evaluate the limit indicated or prove that it does not exist.
1. $\displaystyle\lim_{x \to 3} \frac{\sqrt{3x}-3}{x-3}$
2. $\displaystyle\lim_{x \to \infty} \frac{3x^4+6x^3-12x+3}{(2x+1)(5x^2+4x+16)(x-2)}$
3. $\displaystyle\lim_{x \to 0} \frac{3x^4 - 2x^3 + x^2 + 10x}{(4x^2-x)(x^5+x^3+x+2)}$
4. $\displaystyle\lim_{x\to0^+}\displaystyle\lim_{y \to 0} x^y$
5. $\displaystyle\lim_{y\to0}\displaystyle\lim_{x \to 0^+} x^y$
2. 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.
1. $f+g$
2. $fg$
3. $h\circ f$
4. $\frac1{f}$, assuming $0 \notin \color{red}{f(S)}$
1. 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 $-\infty$.
2. 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 $-\infty$.
3. 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 $y \in \mathbb{R}$.
4. Evaluate the following limits:
1. $\displaystyle\lim_{x\to1}\frac{x^3-1}{x-1}$
2. $\displaystyle\lim_{x\to1}\frac{x^n-1}{x-1}$, for $n \in \mathbb{N}$
3. $\displaystyle\lim_{x\to1}\frac{x^n-x^m}{x-1}$, for $n, m \in \mathbb{N}$
5. Bonus:

A set $S$ is said to be countable if there is some map $f : S \to \mathbb{N}$ which is one-to-one. 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.