l3lab/ntp-mathlib huggingface dataset:
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | 

Proof state:
case hdf
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] a, DifferentiableAt ℝ (fun x => f x) x

Next tactic:
apply eventually_nhdsWithin_of_eventually_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (

Proof state:
case hg'
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] a, deriv (fun x => g x) x ≠ 0

Next tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | 

Proof state:
case hg'
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] a, deriv (fun x => g x) x ≠ 0

Next tactic:
apply eventually_nhdsWithin_of_eventually_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (

Proof state:
case hfa
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => f x) (𝓝[≠] a) (𝓝 0)

Next tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | 

Proof state:
case hfa
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => f x) (𝓝[≠] a) (𝓝 0)

Next tactic:
apply eventually_nhdsWithin_of_eventually_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      

Proof state:
case hfa
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => f x) (𝓝[≠] a) (𝓝 0)

Next tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (

Proof state:
case hga
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => g x) (𝓝[≠] a) (𝓝 0)

Next tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | 

Proof state:
case hga
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => g x) (𝓝[≠] a) (𝓝 0)

Next tactic:
apply eventually_nhdsWithin_of_eventually_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      

Proof state:
case hga
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => g x) (𝓝[≠] a) (𝓝 0)

Next tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (

Proof state:
case hdiv
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (𝓝[≠] a) l

Next tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | 

Proof state:
case hdiv
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (𝓝[≠] a) l

Next tactic:
apply eventually_nhdsWithin_of_eventually_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      

Proof state:
case hdiv
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (𝓝[≠] a) l

Next tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds) <;> 

Proof state:
case hdf.h
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ (fun x => f x) x

Next tactic:
assumption
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds) <;> 

Proof state:
case hg'.h
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝 a, deriv (fun x => g x) x ≠ 0

Next tactic:
assumption
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds) <;> 

Proof state:
case hfa.h
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => f x) (𝓝 a) (𝓝 0)

Next tactic:
assumption
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds) <;> 

Proof state:
case hga.h
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => g x) (𝓝 a) (𝓝 0)

Next tactic:
assumption
/-- **L'Hôpital's rule** for approaching a real, `deriv` version -/
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l := by
  apply lhopital_zero_nhds' <;>
    (first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds) <;> 

Proof state:
case hdiv.h
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (𝓝 a) l

Next tactic:
assumption
/-- **L'Hôpital's rule** for approaching +∞, `deriv` version -/
theorem lhopital_zero_atTop (hdf : ∀ᶠ x : ℝ in atTop, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atTop, deriv g x ≠ 0) (hftop : Tendsto f atTop (𝓝 0))
    (hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :
    Tendsto (fun x => f x / g x) atTop l := by
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l
⊢ Tendsto (fun x => f x / g x) atTop l

Next tactic:
have hdg : ∀ᶠ x in atTop, DifferentiableAt ℝ g x := hg'.mp
    (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
/-- **L'Hôpital's rule** for approaching +∞, `deriv` version -/
theorem lhopital_zero_atTop (hdf : ∀ᶠ x : ℝ in atTop, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atTop, deriv g x ≠ 0) (hftop : Tendsto f atTop (𝓝 0))
    (hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :
    Tendsto (fun x => f x / g x) atTop l := by
  have hdg : ∀ᶠ x in atTop, DifferentiableAt ℝ g x := hg'.mp
    (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l
hdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x
⊢ Tendsto (fun x => f x / g x) atTop l

Next tactic:
have hdf' : ∀ᶠ x in atTop, HasDerivAt f (deriv f x) x :=
    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
/-- **L'Hôpital's rule** for approaching +∞, `deriv` version -/
theorem lhopital_zero_atTop (hdf : ∀ᶠ x : ℝ in atTop, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atTop, deriv g x ≠ 0) (hftop : Tendsto f atTop (𝓝 0))
    (hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :
    Tendsto (fun x => f x / g x) atTop l := by
  have hdg : ∀ᶠ x in atTop, DifferentiableAt ℝ g x := hg'.mp
    (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
  have hdf' : ∀ᶠ x in atTop, HasDerivAt f (deriv f x) x :=
    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l
hdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (deriv f x) x
⊢ Tendsto (fun x => f x / g x) atTop l

Next tactic:
have hdg' : ∀ᶠ x in atTop, HasDerivAt g (deriv g x) x :=
    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
/-- **L'Hôpital's rule** for approaching +∞, `deriv` version -/
theorem lhopital_zero_atTop (hdf : ∀ᶠ x : ℝ in atTop, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atTop, deriv g x ≠ 0) (hftop : Tendsto f atTop (𝓝 0))
    (hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :
    Tendsto (fun x => f x / g x) atTop l := by
  have hdg : ∀ᶠ x in atTop, DifferentiableAt ℝ g x := hg'.mp
    (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
  have hdf' : ∀ᶠ x in atTop, HasDerivAt f (deriv f x) x :=
    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
  have hdg' : ∀ᶠ x in atTop, HasDerivAt g (deriv g x) x :=
    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l
hdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (deriv f x) x
hdg' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt g (deriv g x) x
⊢ Tendsto (fun x => f x / g x) atTop l

Next tactic:
exact HasDerivAt.lhopital_zero_atTop hdf' hdg' hg' hftop hgtop hdiv
/-- **L'Hôpital's rule** for approaching -∞, `deriv` version -/
theorem lhopital_zero_atBot (hdf : ∀ᶠ x : ℝ in atBot, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atBot, deriv g x ≠ 0) (hfbot : Tendsto f atBot (𝓝 0))
    (hgbot : Tendsto g atBot (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :
    Tendsto (fun x => f x / g x) atBot l := by
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atBot, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
have hdg : ∀ᶠ x in atBot, DifferentiableAt ℝ g x :=
    hg'.mp (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
/-- **L'Hôpital's rule** for approaching -∞, `deriv` version -/
theorem lhopital_zero_atBot (hdf : ∀ᶠ x : ℝ in atBot, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atBot, deriv g x ≠ 0) (hfbot : Tendsto f atBot (𝓝 0))
    (hgbot : Tendsto g atBot (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :
    Tendsto (fun x => f x / g x) atBot l := by
  have hdg : ∀ᶠ x in atBot, DifferentiableAt ℝ g x :=
    hg'.mp (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atBot, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l
hdg : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ g x
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
have hdf' : ∀ᶠ x in atBot, HasDerivAt f (deriv f x) x :=
    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
/-- **L'Hôpital's rule** for approaching -∞, `deriv` version -/
theorem lhopital_zero_atBot (hdf : ∀ᶠ x : ℝ in atBot, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atBot, deriv g x ≠ 0) (hfbot : Tendsto f atBot (𝓝 0))
    (hgbot : Tendsto g atBot (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :
    Tendsto (fun x => f x / g x) atBot l := by
  have hdg : ∀ᶠ x in atBot, DifferentiableAt ℝ g x :=
    hg'.mp (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
  have hdf' : ∀ᶠ x in atBot, HasDerivAt f (deriv f x) x :=
    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atBot, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l
hdg : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt f (deriv f x) x
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
have hdg' : ∀ᶠ x in atBot, HasDerivAt g (deriv g x) x :=
    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
/-- **L'Hôpital's rule** for approaching -∞, `deriv` version -/
theorem lhopital_zero_atBot (hdf : ∀ᶠ x : ℝ in atBot, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atBot, deriv g x ≠ 0) (hfbot : Tendsto f atBot (𝓝 0))
    (hgbot : Tendsto g atBot (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :
    Tendsto (fun x => f x / g x) atBot l := by
  have hdg : ∀ᶠ x in atBot, DifferentiableAt ℝ g x :=
    hg'.mp (eventually_of_forall fun _ hg' =>
      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
  have hdf' : ∀ᶠ x in atBot, HasDerivAt f (deriv f x) x :=
    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
  have hdg' : ∀ᶠ x in atBot, HasDerivAt g (deriv g x) x :=
    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atBot, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l
hdg : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt f (deriv f x) x
hdg' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt g (deriv g x) x
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
exact HasDerivAt.lhopital_zero_atBot hdf' hdg' hg' hfbot hgbot hdiv