pow_div_pow_eventuallyEq_atTop — Mathlib

English

For natural p, q with q < p, the quotient x^p / x^q is eventually equal to x^(p−q) as x → ∞.

Русский

При натуральных p, q с q < p, отношение x^p / x^q дробно равно x^(p−q) для sufficiently больших x.

LaTeX

$$$\bigl(\lambda x. x^p / x^q\bigr) =^\text{ᵉɡ}_{atTop} \bigl(\lambda x. x^{p-q}\bigr)$$$

Lean4

theorem pow_div_pow_eventuallyEq_atTop {p q : ℕ} : (fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atTop] fun x => x ^ ((p : ℤ) - q) :=
  by
  apply (eventually_gt_atTop (0 : 𝕜)).mono fun x hx => _
  intro x hx
  simp [zpow_sub₀ hx.ne']

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