File: Mathlib/Analysis/Calculus/LHopital.lean

full_name: HasDerivAt.lhopital_zero_right_on_Ioo
code: theorem lhopital_zero_right_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l

full_name: HasDerivAt.lhopital_zero_right_on_Ico
code: theorem lhopital_zero_right_on_Ico (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ico a b))
    (hcg : ContinuousOn g (Ico a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfa : f a = 0) (hga : g a = 0)
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l

full_name: HasDerivAt.lhopital_zero_left_on_Ioo
code: theorem lhopital_zero_left_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
    (hfb : Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Tendsto g (𝓝[<] b) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) :
  Tendsto (fun x => f x / g x) (𝓝[<] b) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] b) l

full_name: HasDerivAt.lhopital_zero_left_on_Ioc
code: theorem lhopital_zero_left_on_Ioc (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ioc a b))
    (hcg : ContinuousOn g (Ioc a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfb : f b = 0) (hgb : g b = 0)
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) :
    Tendsto (fun x => f x / g x) (𝓝[<] b) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] b) l

full_name: HasDerivAt.lhopital_zero_atTop_on_Ioi
code: 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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l

full_name: HasDerivAt.lhopital_zero_atBot_on_Iio
code: 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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l

full_name: deriv.lhopital_zero_right_on_Ioo
code: theorem lhopital_zero_right_on_Ioo (hdf : DifferentiableOn ℝ f (Ioo a b))
    (hg' : ∀ x ∈ Ioo a b, 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 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l

full_name: deriv.lhopital_zero_right_on_Ico
code: theorem lhopital_zero_right_on_Ico (hdf : DifferentiableOn ℝ f (Ioo a b))
    (hcf : ContinuousOn f (Ico a b)) (hcg : ContinuousOn g (Ico a b))
    (hg' : ∀ x ∈ Ioo a b, (deriv g) x ≠ 0) (hfa : f a = 0) (hga : g a = 0)
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l

full_name: deriv.lhopital_zero_left_on_Ioo
code: theorem lhopital_zero_left_on_Ioo (hdf : DifferentiableOn ℝ f (Ioo a b))
    (hg' : ∀ x ∈ Ioo a b, (deriv g) x ≠ 0) (hfb : Tendsto f (𝓝[<] b) (𝓝 0))
    (hgb : Tendsto g (𝓝[<] b) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[<] b) l) :
    Tendsto (fun x => f x / g x) (𝓝[<] b) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] b) l

full_name: deriv.lhopital_zero_atTop_on_Ioi
code: theorem lhopital_zero_atTop_on_Ioi (hdf : DifferentiableOn ℝ f (Ioi a))
    (hg' : ∀ x ∈ Ioi a, (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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l

full_name: deriv.lhopital_zero_atBot_on_Iio
code: theorem lhopital_zero_atBot_on_Iio (hdf : DifferentiableOn ℝ f (Iio a))
    (hg' : ∀ x ∈ Iio a, (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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l

full_name: HasDerivAt.lhopital_zero_nhds_right
code: /-- L'Hôpital's rule for approaching a real from the right, `HasDerivAt` version -/
theorem lhopital_zero_nhds_right (hff' : ∀ᶠ x in 𝓝[>] a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝[>] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝[>] a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l

full_name: HasDerivAt.lhopital_zero_nhds_left
code: /-- L'Hôpital's rule for approaching a real from the left, `HasDerivAt` version -/
theorem lhopital_zero_nhds_left (hff' : ∀ᶠ x in 𝓝[<] a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝[<] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝[<] a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[<] a) (𝓝 0)) (hga : Tendsto g (𝓝[<] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[<] a) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] a) l

full_name: HasDerivAt.lhopital_zero_nhds'
code: /-- L'Hôpital's rule for approaching a real, `HasDerivAt` version. This
  does not require anything about the situation at `a` -/
theorem lhopital_zero_nhds' (hff' : ∀ᶠ x in 𝓝[≠] a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝[≠] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝[≠] a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[≠] a) (𝓝 0)) (hga : Tendsto g (𝓝[≠] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[≠] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l

full_name: HasDerivAt.lhopital_zero_nhds
code: /-- **L'Hôpital's rule** for approaching a real, `HasDerivAt` version -/
theorem lhopital_zero_nhds (hff' : ∀ᶠ x in 𝓝 a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝 a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝 a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝 a) l) : Tendsto (fun x => f x / g x) (𝓝[≠] a) l 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l

full_name: HasDerivAt.lhopital_zero_atTop
code: /-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, 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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l

full_name: HasDerivAt.lhopital_zero_atBot
code: /-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, 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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l

full_name: deriv.lhopital_zero_nhds_right
code: /-- **L'Hôpital's rule** for approaching a real from the right, `deriv` version -/
theorem lhopital_zero_nhds_right (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 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l

full_name: deriv.lhopital_zero_nhds_left
code: /-- **L'Hôpital's rule** for approaching a real from the left, `deriv` version -/
theorem lhopital_zero_nhds_left (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 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] a) l

full_name: deriv.lhopital_zero_nhds'
code: /-- **L'Hôpital's rule** for approaching a real, `deriv` version. This
  does not require anything about the situation at `a` -/
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 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l

full_name: deriv.lhopital_zero_nhds
code: /-- **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 
goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l

full_name: deriv.lhopital_zero_atTop
code: /-- **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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l

full_name: deriv.lhopital_zero_atBot
code: /-- **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 
goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l