case inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
c d : 𝕜
⊢ Tendsto (fun x => c * x ^ 0) atTop (𝓝 d) ↔ (c = 0 ∨ 0 = 0) ∧ c = d

case inr
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
rcases eq_or_ne n 0 with (rfl | hn)
case inr.inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : c < 0
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d

case inr.inr.inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x => 0 * x ^ n) atTop (𝓝 d) ↔ (0 = 0 ∨ n = 0) ∧ 0 = d

case inr.inr.inr
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
case inr
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
rcases lt_trichotomy c 0 with (hc | rfl | hc)
no goals
case inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
c d : 𝕜
⊢ Tendsto (fun x => c * x ^ 0) atTop (𝓝 d) ↔ (c = 0 ∨ 0 = 0) ∧ c = d
simp [tendsto_const_nhds_iff]
case inr.inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : c < 0
this : Tendsto (fun x => c * x ^ n) atTop atBot
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
case inr.inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : c < 0
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
have := tendsto_const_mul_pow_atBot_iff.2 ⟨hn, hc⟩
no goals
case inr.inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : c < 0
this : Tendsto (fun x => c * x ^ n) atTop atBot
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
simp [not_tendsto_nhds_of_tendsto_atBot this, hc.ne, hn]
no goals
case inr.inr.inl
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x => 0 * x ^ n) atTop (𝓝 d) ↔ (0 = 0 ∨ n = 0) ∧ 0 = d
simp [tendsto_const_nhds_iff]
case inr.inr.inr
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
this : Tendsto (fun x => c * x ^ n) atTop atTop
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
case inr.inr.inr
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
have := tendsto_const_mul_pow_atTop_iff.2 ⟨hn, hc⟩
no goals
case inr.inr.inr
𝕜 : Type u_1
α : Type ?u.36297
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
this : Tendsto (fun x => c * x ^ n) atTop atTop
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
simp [not_tendsto_nhds_of_tendsto_atTop this, hc.ne', hn]