Is $\lim_{n \to \infty}\frac{a}{\frac{b}{n}}$ equal to $\infty$ or undefined? Where $a$ and $b$ are constants. 
I can think of it two different ways. First is that as $n$ goes to infinity, $\frac{b}{n}$ goes to $0$, so that we end up with $\lim_{n \to \infty}\frac{a}{\frac{b}{n}} = \frac{a}{0}$, which is undefined.
The other way is to say that $$\lim_{n \to \infty}\frac{a}{\frac{b}{n}} = \lim_{n \to \infty} a \cdot\frac{n}{b} = a \cdot \infty = \infty$$.
Which one is correct and why?

Well,
$$\lim_{n \to \infty} f(n)g(n) = \left( \lim_{n \to \infty} f(n) \right) \left(\lim_{n \to \infty} g(n) \right)$$ only holds if both $\lim_{n \to \infty} f(n)$ and $\lim_{n \to \infty} g(n)$ exist.
In this case, $f(n)=a$ and $g(n) = \frac{n}{b}$. It is clear then that $\lim_{n \to \infty} g(n) $ does not exist.
