R : Type u_1
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : Rˣ
⊢ (fun x_1 => inverse (1 + ↑x⁻¹ * x_1) * ↑x⁻¹) =O[𝓝 0] fun _t => 1 * 1
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : Rˣ
⊢ (fun t => inverse (↑x + t)) =O[𝓝 0] fun _t => 1
refine EventuallyEq.trans_isBigO (inverse_add x) (one_mul (1 : ℝ) ▸ ?_)
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : Rˣ
⊢ (fun x_1 => inverse (1 - -↑x⁻¹ * x_1) * ↑x⁻¹) =O[𝓝 0] fun _t => 1 * 1
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : Rˣ
⊢ (fun x_1 => inverse (1 + ↑x⁻¹ * x_1) * ↑x⁻¹) =O[𝓝 0] fun _t => 1 * 1
simp only [← sub_neg_eq_add, ← neg_mul]
no goals
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : Rˣ
hzero : Tendsto (fun x_1 => -↑x⁻¹ * x_1) (𝓝 0) (𝓝 0)
⊢ (fun x_1 => inverse (1 - -↑x⁻¹ * x_1) * ↑x⁻¹) =O[𝓝 0] fun _t => 1 * 1
exact (inverse_one_sub_norm.comp_tendsto hzero).mul (isBigO_const_const _ one_ne_zero _)