Bayer and Mumford, What can be computed in algebraic geometry, reads (in part):
Let $S = k[x_0, \ldots, x_n]$ be the homogeneous coordinate ring of $\mathbb{P}^n$.
[. . .]
Choose a one-parameter subgroup $\lambda(t) < GL_{n+1}$ of the form $$\lambda(t) = \begin{bmatrix}t^{w_0} \\ & t^{w_1} \\ & & \ddots \\ & & & t^{w_n} \end{bmatrix}$$ where $W = (w_0, \ldots, w_n)$ is a vector of integer weights.
[. . .]
Let $\lambda$ act on $S$ by mapping each $x_i$ to $t^{w_i} x_i$.
[. . .]
We take the projective limit $in(f) = \lim_{t \to 0} \lambda f$ by collecting the terms of $\lambda f$ involving the least power of $t$.
I'm not clear on this last sentence. Are they defining the term "projective limit" and the notation $\lim_{t \to 0}$ here? Or is there a general notion of a projective limit of -- polynomials? ring elements? -- that is being used here?