TITLE: Can I get a sequence of bounded functions converging pointwise to $f(x)=1/x$ for $x$ non zero and $0$ for $x$ zero?
QUESTION [6 upvotes]: How do I construct an explicit sequence of bounded functions converging pointwise to $f(x)=1/x$ for non zero $x$ and $f(0)=0$. It would be better if someone may find a continuous and even better if someone gives a differentiable sequence of functions on $\Bbb R$ converging to $f$ pointwise.

REPLY [11 votes]: The functions
$$
f_n(x) = \frac{nx}{1+nx^2}
$$
are (infinitely often) differentiable on $\Bbb R$. Each $f_n$ is bounded:
$$
 |f_n(x)| =  \frac{\sqrt{nx^2}}{1+nx^2} \sqrt n \le \frac 12 \sqrt n \, ,
$$
and $f_n(x) \to f(x)$ pointwise for all $x$.
Graph of $f_5, f_{10}, f_{40}$ together with the graph of $1/x$: