Limits and While Loops 
Question: Consider the following program. Does $f(1)=\infty$? 
\begin{align*}
f(i):=&|\text{while } \frac{1}{i}>0\\
&||i\leftarrow i+1\\
&|i
\end{align*}

I would say that $f(1)=\infty$ is a true statement. The program does not terminate, but one could consider the sequence of points $\{x_i\}_{i\in \mathbb{N}}$ given by $x_i:=\frac{1}{i}$, so that  $\{x_i\}_{i\in \mathbb{N}}$ converges to the limit $0$ which makes $\underbrace{\frac{1}{i}}_{=0}>0$ false which means $f(1)=\infty$. However, I could be wrong.

The only thing we can say here is that $f(1)$ is not defined, or that the program does not terminate for the input $1$.
We cannot say that $f(1)=0$.
