Theorem: Polynomial.div_tendsto_zero_of_degree_lt
File: Mathlib/Analysis/SpecialFunctions/Polynomials.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
⊢ Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0)
state_after: case pos
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
hP : P = 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0)

case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
hP : ¬P = 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0)
tactic: by_cases hP : P = 0
state_before: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
hP : ¬P = 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0)
state_after: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0)
tactic: rw [← natDegree_lt_natDegree_iff hP] at hdeg
state_before: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0)
state_after: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop (𝓝 0)
tactic: refine (isEquivalent_atTop_div P Q).symm.tendsto_nhds ?_
state_before: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop (𝓝 0)
state_after: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop (𝓝 (?neg✝ * 0))

case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ 𝕜
tactic: rw [← mul_zero]
state_before: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop (𝓝 (?neg✝ * 0))

case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ 𝕜
state_after: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ ↑P.natDegree - ↑Q.natDegree < 0
tactic: refine (tendsto_zpow_atTop_zero ?_).const_mul _
state_before: case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.natDegree < Q.natDegree
hP : ¬P = 0
⊢ ↑P.natDegree - ↑Q.natDegree < 0
state_after: no goals
tactic: omega