What's the projective limit of these polynomial rings ? Define an inverse system of polynomial rings over a commutative ring $k$ by the canonical projection $k[x_1,...,x_n] \to k[x_1,...,x_m]\;(m&lt; n)$. 

Question: What is the projective limit $\varprojlim_n k[x_1,...,x_n]$ ?
Note that we can write, as sets:

$$ k[x_1, x_2, \cdots, x_n] \cong k[x_1] \times x_2 k[x_1, x_2] \times \cdots \times x_n k[x_1, x_2, \cdots, x_n]$$

(all but the first component of the product are ideals) An element of the product is interpreted as the sum of all of its components. Furthermore, the canonical projection (assuming you mean to send $x_n$ to $0$) is just the projections onto the first $m$ components.

Therefore, 

$$ \lim_n k[x_1, x_2, \cdots, x_n] \cong k[x_1] \times x_2 k[x_1, x_2] \times x_3 k[x_1, x_2, x_3] \times \cdots $$

What addition is should be straightforward. Working with multiplication should be similar in flavor to working in power series rings.

The elements should probably be thought of as infinite sums; given a finite set of variables, each such sum should only have finitely many monomials that involve only those variables.
I can answer your question. Call a linear combination of monomials $\prod_i x_i^{m_i}$ such that $m_i\ge 0, \sum_i m_i=m$ an $m$-form. Then as a subring 
$$\varprojlim_{n \to \infty} k[x_1,...,x_n] = \lbrace f_1 + \cdots + f_m \mid m \ge 0,\;\; f_i\; i\text{-form}\rbrace \le k[[x_1,x_2,...]]$$

For the proof note that there are canonical projections $p_n: k[[x_1,x_2,...]] \to k[[x_1,...,x_n]]$ and the restriction to $S = \lbrace f_1 + \cdots + f_m \mid m \ge 0,\;\; f_i\; i\text{-form}\rbrace$ has image in $k[x_1,...,x_n]$. Hence $p= \prod_n p_n: S \to \prod_n k[x_1,...,x_n]$ is an embedding of rings whose image is exactly $\varprojlim_{n \to \infty} k[x_1,...,x_n]$.