Use this tag for questions related to the analytic spread of an ideal of a local ring, which is the Krull dimension of the fiber at the special point of the local ring of the Rees algebra of the ideal.

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### asymptotic function of function graph: $y=\frac{{{x}^{x+1}}}{{{(x+1)}^{x}}}$ [closed]

Find the asymptotic function of the function graph: $$y=\frac{{{x}^{x+1}}}{{{(x+1)}^{x}}}$$ Please, help. Thank you.
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### Evaluating $\lim _{x\to \infty }\left(\frac{7x^2-1}{7x^2+5}\right)^x$ [closed]

How can I find the limits of the following sequence? $$\lim _{x\to \infty }\left(\frac{7x^2-1}{7x^2+5}\right)^x$$
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### floor function: $\lfloor ax\rfloor-a\lfloor x\rfloor ≤ k$

$\forall a \in \mathbb{N}, \exists k \in \mathbb{N}$ such that $$\forall x \in \mathbb{R}, \;\lfloor ax\rfloor-a\lfloor x\rfloor \le k$$ I don't have much experience proving the floor or ceiling so, ...
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### how does Macaulay2 computes analytic spread for non-local rings?

Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring. In the books like ...
Let $(R,\mathbb{m})$ be a Noetherian local ring and $I$ an ideal of $R$. Let $t$ be an indeterminate over $R$. The analytic spread $l(I)$ of $I$ is defined to be the Krull dimension of the ring \$R[It]/...