l3lab/ntp-mathlib huggingface dataset:
theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  obtain ⟨a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' := ⟨1 + max a 0,
    ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
      lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
  have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
  have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)
  have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hff' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hgg' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have := lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng
    (by
      intro x hx
      refine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)
      exact hg' _ (fact2 x hx))
    (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop)
    (by
      refine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)
      rw [eventuallyEq_iff_exists_mem]
      use Ioi 0, self_mem_nhdsWithin
      intro x hx
      unfold Function.comp
      simp only
      

Proof state:
case right
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
x : ℝ
hx : x ∈ Ioi 0
⊢ f' x⁻¹ / g' x⁻¹ = f' x⁻¹ * -(x ^ 2)⁻¹ / (g' x⁻¹ * -(x ^ 2)⁻¹)

Next tactic:
erw [mul_div_mul_right]
theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  obtain ⟨a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' := ⟨1 + max a 0,
    ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
      lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
  have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
  have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)
  have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hff' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hgg' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have := lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng
    (by
      intro x hx
      refine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)
      exact hg' _ (fact2 x hx))
    (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop)
    (by
      refine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)
      rw [eventuallyEq_iff_exists_mem]
      use Ioi 0, self_mem_nhdsWithin
      intro x hx
      unfold Function.comp
      simp only
      erw [mul_div_mul_right]
      

Proof state:
case right.hc
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
x : ℝ
hx : x ∈ Ioi 0
⊢ -(x ^ 2)⁻¹ ≠ 0

Next tactic:
refine' neg_ne_zero.mpr (inv_ne_zero <| pow_ne_zero _ <| ne_of_gt hx)
theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  obtain ⟨a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' := ⟨1 + max a 0,
    ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
      lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
  have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
  have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)
  have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hff' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hgg' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have := lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng
    (by
      intro x hx
      refine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)
      exact hg' _ (fact2 x hx))
    (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop)
    (by
      refine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)
      rw [eventuallyEq_iff_exists_mem]
      use Ioi 0, self_mem_nhdsWithin
      intro x hx
      unfold Function.comp
      simp only
      erw [mul_div_mul_right]
      refine' neg_ne_zero.mpr (inv_ne_zero <| pow_ne_zero _ <| ne_of_gt hx))
  

Proof state:
case intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
this : Tendsto (fun x => (f ∘ Inv.inv) x / (g ∘ Inv.inv) x) (𝓝[>] 0) l
⊢ Tendsto (fun x => f x / g x) atTop l

Next tactic:
have := this.comp tendsto_inv_atTop_zero'
theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  obtain ⟨a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' := ⟨1 + max a 0,
    ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
      lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
  have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
  have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)
  have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hff' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hgg' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have := lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng
    (by
      intro x hx
      refine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)
      exact hg' _ (fact2 x hx))
    (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop)
    (by
      refine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)
      rw [eventuallyEq_iff_exists_mem]
      use Ioi 0, self_mem_nhdsWithin
      intro x hx
      unfold Function.comp
      simp only
      erw [mul_div_mul_right]
      refine' neg_ne_zero.mpr (inv_ne_zero <| pow_ne_zero _ <| ne_of_gt hx))
  have := this.comp tendsto_inv_atTop_zero'
  

Proof state:
case intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
this✝ : Tendsto (fun x => (f ∘ Inv.inv) x / (g ∘ Inv.inv) x) (𝓝[>] 0) l
this : Tendsto ((fun x => (f ∘ Inv.inv) x / (g ∘ Inv.inv) x) ∘ fun r => r⁻¹) atTop l
⊢ Tendsto (fun x => f x / g x) atTop l

Next tactic:
unfold Function.comp at this
theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  obtain ⟨a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' := ⟨1 + max a 0,
    ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
      lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
  have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
  have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)
  have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hff' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
    comp x (hgg' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
  have := lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng
    (by
      intro x hx
      refine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)
      exact hg' _ (fact2 x hx))
    (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop)
    (by
      refine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)
      rw [eventuallyEq_iff_exists_mem]
      use Ioi 0, self_mem_nhdsWithin
      intro x hx
      unfold Function.comp
      simp only
      erw [mul_div_mul_right]
      refine' neg_ne_zero.mpr (inv_ne_zero <| pow_ne_zero _ <| ne_of_gt hx))
  have := this.comp tendsto_inv_atTop_zero'
  unfold Function.comp at this
  

Proof state:
case intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
this✝ : Tendsto (fun x => (f ∘ Inv.inv) x / (g ∘ Inv.inv) x) (𝓝[>] 0) l
this : Tendsto (fun x => (fun x => f x⁻¹ / g x⁻¹) ((fun r => r⁻¹) x)) atTop l
⊢ Tendsto (fun x => f x / g x) atTop l

Next tactic:
simpa only [inv_inv]
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rw [preimage_neg_Iio] at hdnf
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  rw [preimage_neg_Iio] at hdnf
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rw [preimage_neg_Iio] at hdng
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  rw [preimage_neg_Iio] at hdnf
  rw [preimage_neg_Iio] at hdng
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
have := lhopital_zero_atTop_on_Ioi hdnf hdng
    (by
      intro x hx h
      apply hg' _ (by rw [← preimage_neg_Iio] at hx; exact hx)
      rwa [mul_comm, ← neg_eq_neg_one_mul, neg_eq_zero] at h)
    (hfbot.comp tendsto_neg_atTop_atBot) (hgbot.comp tendsto_neg_atTop_atBot)
    (by
      simp only [mul_one, mul_neg, neg_div_neg_eq]
      exact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_atTop_atBot))
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  rw [preimage_neg_Iio] at hdnf
  rw [preimage_neg_Iio] at hdng
  have := lhopital_zero_atTop_on_Ioi hdnf hdng
    (by
      

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
⊢ ∀ x ∈ Ioi (-a), g' (-x) * -1 ≠ 0

Next tactic:
intro x hx h
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  rw [preimage_neg_Iio] at hdnf
  rw [preimage_neg_Iio] at hdng
  have := lhopital_zero_atTop_on_Ioi hdnf hdng
    (by
      intro x hx h
      

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
x : ℝ
hx : x ∈ Ioi (-a)
h : g' (-x) * -1 = 0
⊢ False

Next tactic:
apply hg' _ (by rw [← preimage_neg_Iio] at hx; exact hx)
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  rw [preimage_neg_Iio] at hdnf
  rw [preimage_neg_Iio] at hdng
  have := lhopital_zero_atTop_on_Ioi hdnf hdng
    (by
      intro x hx h
      apply hg' _ (by 

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
x : ℝ
hx : x ∈ Ioi (-a)
h : g' (-x) * -1 = 0
⊢ ?m.32398 ∈ Iio a

Next tactic:
rw [← preimage_neg_Iio] at hx
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  rw [preimage_neg_Iio] at hdnf
  rw [preimage_neg_Iio] at hdng
  have := lhopital_zero_atTop_on_Ioi hdnf hdng
    (by
      intro x hx h
      apply hg' _ (by rw [← preimage_neg_Iio] at hx; 

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
x : ℝ
hx✝ : x ∈ Ioi (-a)
hx : x ∈ -Iio a
h : g' (-x) * -1 = 0
⊢ ?m.32487 h ∈ Iio a

Next tactic:
exact hx
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
  have hdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
    comp x (hff' (-x) hx) (hasDerivAt_neg x)
  have hdng : ∀ x ∈ -Iio a, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
    comp x (hgg' (-x) hx) (hasDerivAt_neg x)
  rw [preimage_neg_Iio] at hdnf
  rw [preimage_neg_Iio] at hdng
  have := lhopital_zero_atTop_on_Ioi hdnf hdng
    (by
      intro x hx h
      apply hg' _ (by rw [← preimage_neg_Iio] at hx; exact hx)
      

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
x : ℝ
hx : x ∈ Ioi (-a)
h : g' (-x) * -1 = 0
⊢ g' (-x) = 0

Next tactic:
rwa [mul_comm, ← neg_eq_neg_one_mul, neg_eq_zero] at h