# Why does this equation converge to 1?

The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow \mathfrak{R}^1$). This equation has the property that the answer will converge to 1, as the input elements become more and more alike.

Anyway, here is the equation. $\bf x$ is the N-dimensional input vector. $y$ is the 1-dimensional scalar output. $e$ is just the exponent operator.

$$y = \frac{e^{\frac{1}{N}\displaystyle\sum_{n=1}^N \log_e(x[n])}}{\frac{1}{N}\displaystyle\sum_{n=0}^N x[n]}$$

(Notice how the denominator is just the mean of $\bf x$). This function will return $y=1$ if all the elements of the vector $x$ are equal to each other.

I am trying to do two things:

1) First, I would like to 'translate' the English statement "Show that this equation converges to 1, as all the elements in the vector $\bf x$ become more and more alike", into a mathematical statement. (Italics stressing what I want translated). For example, in English we might say "As the variable $c$ approaches infinity, etc, and we use the $\displaystyle\lim_{c \to +\infty}$ to mathematically denote that.

2) After that, I would like to show (prove), that the above equation actually does converge to 1, as the elements of the above equation $x$ become more and more similar. (I know that it does indeed converge to 1, but would like to prove it).

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First of all, note that in order to get real values, you want all $x_n > 0$. The equation can be simplified as $$y = \dfrac{\prod_{n=1}^N x_n^{1/N}}{\frac{1}{N} \sum_{n=1}^N x_n}$$ You could say that for any $c > 0$, $$\lim_{{\bf x} \to [c,\ldots,c]} \dfrac{\prod_{n=1}^N x_n^{1/N}}{\frac{1}{N} \sum_{n=1}^N x_n} = 1$$ This is true because the expression is a continuous function on $(0,\infty)^N$ and, as you noted, the value is $1$ when all $x_n$ are equal to the same positive number.
Thank you for the expression in the limit. However, I am failing to see how/why it being a continuous function means that it is going to converge to 1, as all elements of $x$ converge to $c$. Could you please expand on that. Thanks. –  Mohammad Jul 23 '12 at 18:11