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18.104
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Spring 2013
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\begin{center}\textbf{Homework 3}\end{center}

\begin{center}Due February 21st by the beginning of class.\end{center}

\textbf{Problem:} Calculate the limits as $n \to \infty$ of the following sequences of distributions on $\mathbb R$:
\[
u_n = \frac{n}{1+n^2x^2}, \quad v_n = \frac{n^3 x}{(1+n^2x^2)^2}, \quad w_n = \frac{n^3 \sin x}{(1+n^2x^2)^2}.
\]

\textit{Hint:} Reduce each limit to the previous one. 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 $\lim_{n \to \infty} \int_{\mathbb R}f_n = \int_{\mathbb R} f$.

\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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