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What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $?

What is the value of $\lim_{n \rightarrow d, d\rightarrow \infty} (n/d)$? What is the function's range? What are the characteristics of any differences in the range given differences specified in magnitude or ordering in the domain?

If it ranges from zero to one and the difference between consecutive elements is zero or infinitesimal, is there any neighborhood in $[0,1]$ without an element of the range? Is it monotone increasing? If it's not increasing, does it range to one?

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I'm not sure I understand the last two paragraphs of your question, but the limits are $$ \lim_{n \to d} \lim_{d \to \infty} \frac{n}{d} = \lim_{n \to d} 0 = 0$$ and $$\lim_{d \to \infty} \lim_{n \to d} \frac{n}{d} = \lim_{d \to \infty} 1 = 1.$$ In general, you can interchange the order of limits. –  JavaMan Jan 21 '13 at 1:57
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I agree with what JavaMan said, except that I'm sure he meant to say that in general, you can't exchange the order of limits. –  mixedmath Jan 21 '13 at 2:16
    
Ross, your word salad is incomprehensible. Please find someone who will help you translate it into mathematics, so that the rest of us might understand it. –  Gerry Myerson Jan 21 '13 at 2:37
    
$$d, m, n > 0 \Rightarrow (m < n \Rightarrow \frac{m}{d} < \frac{n}{d} \Rightarrow f(m) < f(n)) \text{($f$ is monotone increasing)} $$ $$(n_1 - m_1 = n_2 - m_2) \Rightarrow \frac{n_1 - m_1}{d} = \frac{n_2 - m_1}{d} \text{($f$ is constant monotone)}$$ If it goes to one (if the function's range, in the course-of-passage over the function's domain $\mathbb{N}$ has limit $1$), $$f(0) = \frac{0}{d} = 0, \lim_{n = 0 \to d} \frac{n}{d} = 1 \text{($f$ ranges from zero to one)}$$ is there any neighborhood in $\mathbb{R}_{[0,1]}$ not containing an element of the range? –  Ross Finlayson Jan 21 '13 at 4:04
    
Ross, the rationals are dense in the reals. Every non-empty open subset of $[0,1]$ contains infinitely many numbers $n/d$ with $n,d$ positive integers and $n\lt d$. –  Gerry Myerson Jan 21 '13 at 8:25

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