# The result of $k/\infty$ where $k$ is positive natural number while the limit of $k/n$ becomes zero as n approaches $\infty$?

So when $\lim_{n \to \infty} k/n$ where $n$ is any positive natural number, the result is zero. However, when $n$ is divided by $\infty$, this may not be zero, as far as I know (the answer seems to be infinitesimal number). This makes me curious: the limit says that as $n$ approaches $\infty$, it converges to zero, yet when $n$ is indeed $\infty$, it is not zero. So what is going on here?

To summarize, how can $k/n$ where $n=\infty$ be non-zero infinitesimal, yet the limit of $k/n$ as $n$ approaches $\infty$ be zero? Wouldn't the result of limit imply that the former is also zero?

• What exactly is the question? – Anthony Peter Sep 15 '13 at 1:33
• For clarity, it is best to avoid using the word "infinity" in a sentence. – André Nicolas Sep 15 '13 at 1:38
• Edited the question. – diffio Sep 15 '13 at 1:38
• By the way, is there any problem with question? If so, please tell me, as there are downvotes here.. – diffio Sep 15 '13 at 1:40
• $\infty$ is not a number in the usual sense, so what does it mean to divide by $\infty$? – Michael Albanese Sep 15 '13 at 19:11

## 1 Answer

What is going on is simple. You don't talk about limits and about infinitesimals in the same context. The point of infinitesimals is not to use limits.