Is $\lim_{n \to \infty}\frac{a}{\frac{b}{n}}$ equal to $\infty$ or undefined?

Question

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?

Answer 1

However it is written at the outset we are given the sequence $$x_n:={a\,n\over b}\qquad(n\geq1)\ ,$$ with the tacit assumption that $b\ne0$. If $a=0$ then $x_n=0$ for all $n$, hence $\lim_{n\to\infty}x_n=0$. If $a\ne0$ then we all know that the $x_n$ converge to $\infty$ if $ab>0$, and to $-\infty$, if $ab<0$. This means that the sequence is divergent in ${\mathbb R}$. Nevertheless we are entitled to write $$\lim_{n\to\infty}x_n=\infty\quad(ab>0),\qquad \lim_{n\to\infty}x_n=-\infty\quad(ab<0)\ ,$$ meaning that we accept $\pm\infty$ as limiting values, and have verified the corresponding convergence conditions.

Answer 2

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.