feat: asymptotics lemmas (#17394)

From https://github.com/MichaelStollBayreuth/EulerProducts

Co-authored-by: Michael Stoll <[email protected]>

Mathlib/Analysis/Asymptotics/Asymptotics.lean

@@ -1251,6 +1251,9 @@ theorem IsLittleO.trans_tendsto (hfg : f'' =o[l] g'') (hg : Tendsto g'' l (𝓝
     Tendsto f'' l (𝓝 0) :=
   hfg.isBigO.trans_tendsto hg
 
+lemma isLittleO_id_one [One F''] [NeZero (1 : F'')] : (fun x : E'' => x) =o[𝓝 0] (1 : E'' → F'') :=
+  isLittleO_id_const one_ne_zero
+
 /-! ### Multiplication by a constant -/
 
 
@@ -1915,6 +1918,13 @@ theorem isBigO_atTop_iff_eventually_exists_pos {α : Type*}
     f =O[atTop] g ↔ ∀ᶠ n₀ in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖f n‖ ≤ ‖g n‖ := by
   simp_rw [isBigO_iff'', ← exists_prop, Subtype.exists', exists_eventually_atTop]
 
+lemma isBigO_mul_iff_isBigO_div {f g h : α → 𝕜} (hf : ∀ᶠ x in l, f x ≠ 0) :
+    (fun x ↦ f x * g x) =O[l] h ↔ g =O[l] (fun x ↦ h x / f x) := by
+  rw [isBigO_iff', isBigO_iff']
+  refine ⟨fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩⟩ <;>
+  · refine H.congr <| Eventually.mp hf <| Eventually.of_forall fun x hx ↦ ?_
+    rw [norm_mul, norm_div, ← mul_div_assoc, le_div_iff₀' (norm_pos_iff.mpr hx)]
+
 end Asymptotics
 
 open Asymptotics