no goals
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 / g x) (𝓝[{a}ᶜ] a) l
apply lhopital_zero_nhds' <;>
  (first | apply eventually_nhdsWithin_of_eventually_nhds |
    apply tendsto_nhdsWithin_of_tendsto_nhds) <;> assumption
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
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}ᶜ] a) l
first | apply eventually_nhdsWithin_of_eventually_nhds |
 apply tendsto_nhdsWithin_of_tendsto_nhds
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}ᶜ] a) l
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}ᶜ] a) l
apply eventually_nhdsWithin_of_eventually_nhds
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
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}ᶜ] a) l
apply tendsto_nhdsWithin_of_tendsto_nhds