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\title{MTH436 - Homework 5}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 3/20/19.
 
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 \section{Warm-up Questions}
 
 \begin{que}
  Compute the following limits.
  \begin{enumerate}
   \item[(i)] $\lim_{x \to 0} \frac{\tan x}{x}$. \vskip 0.1cm
   \item[(ii)] $\lim_{x \to 0^+} \frac{1}{x(\ln x)^2}$ \vskip 0.1cm
   \item[(iii)] $\lim_{x \to \infty} \left( \frac{1}{x} \right)^x$. \vskip 0.1cm
   \item[(iv)] $\lim_{x \to {\frac{\pi}{2}}^-} (\sec x - \tan x)$. 
   
  \end{enumerate}

 \end{que}

 \begin{que}
  Let $f \colon (a,b) \to \R$ be differentiable and suppose that $f''(x)$ exists at $x \in (a,b)$. Prove that
  \[
   f''(x) = \lim_{h \to 0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}.
   \]
Give an example to show that the limit on the right hand side may exist even if $f''(x)$ does not.
 \end{que}

 \begin{que}
  Let $f$ and $g$ be $n$-times differentiable at $x$. Prove that
  \[
   (fg)^{(n)}(x) = \sum_{k=0}^{n} \frac{n!}{k!(n-k)!} f^{(n-k)}(x)g^{(k)}(x).
  \]

 \end{que}

 \begin{que}
  Suppose $f \colon \R \to \R$ is differentiable. Define $f^{\circ n} = (\overbrace{f \circ f \circ \dotso \circ f}^{\text{$n$-times}})$. Prove that
  \[
   (f^{\circ n})'(x) = \prod_{k=0}^{n-1} f'(f^{\circ k}(x)) = f'(x) \cdot f'(f(x))  \cdots  f'(f^{\circ (n-1)}(x))
  \]
In particular, if we have a finite set $\{ x_1, x_2, \ldots x_n \}$ such that $f(x_i) = x_{i+1}$ for $1 \leq i < n$ and $f(x_n) = x_1$, then $(f^{\circ n})'(x_i) = (f^{\circ n})'(x_j)$ for all $1 \leq i,j \leq n$.
 \end{que}

 \begin{que}
  Let $f \colon \R \to \R$ be differentiable. We say $x$ is a fixed point of $f$ if $f(x) = x$. 
  \begin{enumerate}
   \item[(i)] Prove that if $f'(t) \neq 1$ for all $t \in \R$ then $f$ has at most one fixed point.
   \item[(ii)] Prove that $g(x) = x  + \frac{1}{1 + e^x}$ has $0 < g'(x) < 1$ for all $x$ but $g$ has no fixed point.
  \end{enumerate}
 
 \end{que}
 
 
  \begin{que}
  Let $f \colon [0,\infty) \to \R$ be differentiable and suppose $\lim_{x \to \infty} (f(x) + f'(x)) = \ell$. Prove that $\lim_{x \to \infty} f(x) = \ell$ and $\lim_{x \to \infty} f'(x) = 0$. 
 \end{que}
 
 \begin{que}
  Suppose we know $\lim_{x \to 0} \frac{f(x)}{g(x)} = L$. Is it true that $\lim_{x \to 0} \frac{f'(x)}{g'(x)} = L$? If not, where does the proof break down in the proof of L'Hospital's rule?
 \end{que}

 
 
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 \section{Submitted Questions}
 
 


\begin{que} In this question, we investigate an infinitely differentiable function which is not equal to its Taylor series. Let
  \[
  f(x) = 	\begin{cases}
		  e^{-1/x^2} \quad &\text{if $x \neq 0$,} \\
		  0	\quad		&\text{if $x = 0$}.
		\end{cases}
 \]
 \begin{enumerate}
  \item Prove that for all $k \in \N$ that $\lim_{y \to \infty} \frac{y^k}{e^{y^2}} = 0$. (Use induction on $k$ and L'Hospital's rule). Convince yourself that $\lim_{y \to -\infty} \frac{y^k}{e^{y^2}} = 0$ (no need to prove this, just check that the same argument will work).
  \item Deduce that $\lim_{x \to 0} \frac{e^{-1/x^2}}{x^k} = 0$.
  \item Let $n \in \N \cup \{0\}$. Prove for $x \neq 0$, that $f^{(k)}(x) = f(x)G_k(x)$, where $G_k(x)$ is a rational function.
  \item Prove by induction and part (ii) that $f^{(n)}(0) = 0$ for all $n \in \N \cup \{0\}$. Observe that $f$ is therefore infinitely differentiable.
  \item Prove that if $R_n(x)$ is the Lagrange form of the remainder from Taylor's theorem and $x \neq 0$ then $\lim_{n \to \infty} R_n(x) \neq 0$.
 \end{enumerate}

 \end{que}

 

 
 
 
 
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 \section{Challenge Questions}
 
 
 
 \begin{que}
  Suppose $f \colon [0,\infty) \to \R$ is continuous with $f(0) = 0$ and $f$ is differentiable at $x$ for all $x > 0$. Show that if $f(x)$ is increasing then $\frac{f(x)}{x}$ is increasing for $x \geq 0$.
 \end{que}

 \begin{que}
  Suppose $f \colon [-1,1] \to \R$ is three times differentiable with
  \[
   f(-1) = 0, \quad f(0) = 0, \quad f(1) = 1, \quad f'(0) = 0. 
  \]
Prove that there exists $x \in (-1,1)$ such that $f'''(x) \geq 3$. Show that the equality is obtained if $f(x) = \frac{1}{2}(x^3 + x^2)$. 
 \end{que}

 
 
 
 
 
 
 
 
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