(** Division *)

(* Specifications: *)
Lemma is_lim_seq_div (u v : nat -> R) (l1 l2 l : Rbar) :
is_lim_seq u l1 -> is_lim_seq v l2 -> l2 <> 0 ->
is_Rbar_div l1 l2 l ->
is_lim_seq (fun n => u n / v n) l.
Lemma is_lim_seq_div' (u v : nat -> R) (l1 l2 : R) :
is_lim_seq u l1 -> is_lim_seq v l2 -> l2 <> 0 ->
is_lim_seq (fun n => u n / v n) (l1 / l2).
Lemma ex_lim_seq_div (u v : nat -> R) :
ex_lim_seq u -> ex_lim_seq v -> Lim_seq v <> 0 ->
ex_Rbar_div (Lim_seq u) (Lim_seq v) ->
ex_lim_seq (fun n => u n / v n).
Lemma Lim_seq_div (u v : nat -> R) :
ex_lim_seq u -> ex_lim_seq v -> (Lim_seq v <> 0) ->
ex_Rbar_div (Lim_seq u) (Lim_seq v) ->
Lim_seq (fun n => u n / v n) = Rbar_div (Lim_seq u) (Lim_seq v).
Lemma is_lim_div (f g : R -> R) (x lf lg : Rbar) :
is_lim f x lf -> is_lim g x lg -> lg <> 0 ->
ex_Rbar_div lf lg ->
is_lim (fun y => f y / g y) x (Rbar_div lf lg).
Lemma ex_lim_div (f g : R -> R) (x : Rbar) :
ex_lim f x -> ex_lim g x -> Lim g x <> 0 ->
ex_Rbar_div (Lim f x) (Lim g x) ->
ex_lim (fun y => f y / g y) x.
Lemma Lim_div (f g : R -> R) (x : Rbar) :
ex_lim f x -> ex_lim g x -> Lim g x <> 0 ->
ex_Rbar_div (Lim f x) (Lim g x) ->
Lim (fun y => f y / g y) x = Rbar_div (Lim f x) (Lim g x).