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18.104
\\
Spring 2013
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\begin{center}\textbf{Homework 4}\end{center}

\begin{center}Due February 26th by the beginning of class.\end{center}

\textbf{Problem:} Prove that 
\[
\lim_{n \to \infty} \frac{n^2x}{1+n^2x^2} =  \textrm{p.v.} \frac 1 x
\]
in $\mathscr D'(\mathbb R)$, where $\textrm{p.v.} \frac 1 x$ denotes the principal value distribution as in \cite[page 19]{frjo}. 

\textit{Hint:} Use the fact that 
\[
\langle \textrm{p.v.} \frac 1 x, \phi \rangle = \int_{-1}^1 \frac{\phi(x)- \phi(0)}{x}dx + \int_{\mathbb R \setminus[-1,1]} \frac{\phi(x)}{x}dx.
\]
You may use without proof the dominated convergence theorem, which says that if $f_n(x) \to f(x)$ for every $x \in \mathbb R$, and if $|f_n(x)| \le g(x)$ for every $n \in \mathbb N$ and $x \in \mathbb R$ for some function $g$ with $\int_{\mathbb R} g < \infty$, then $\lim_{n \to \infty} \int_{\mathbb R}f_n = \int_{\mathbb R} f$.

\textit{Not required:} What about $\lim  \frac{n^2 f(x)}{1+n^2x^2}$, where $f \in C^\infty(\mathbb R)$?

\textbf{Solution:}


\begin{thebibliography}{00}
\bibitem[FrJo]{frjo} G. Friedlander and M. Joshi. The Theory of Distributions, second edition, Cambridge University Press, 1998.
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