>>16546882252+ Atk Life Orb Arcanine Extreme Speed vs. 252 HP / 0 Def Talonflame: 147-174 (40.8 - 48.3%) -- guaranteed 3HKO
-1 252+ Atk Talonflame Brave Bird vs. 0 HP / 0 Def Arcanine: 127-150 (39.5 - 46.7%) -- guaranteed 3HKO
Arcanine has 321 HP
Switch in, Intimidate
Talonflame drops to -1 Atk
Talonflame uses Brave Bird
Arcanine loses an average of 138 HP (Drops to 183 HP)
Talonflame loses 46 HP from recoil (Drops to 314)
Arcanine uses ExtremeSpeed
Talonflame loses ~158 HP (Drops to 156 HP)
Arcanine loses 10% of its max health from Life Orb (which is 32 HP, drops to 151 HP)
Talonflame uses Brave Bird
Arcanine loses ~138 HP (drops to 13 HP)
Talonflame loses 46 HP from recoil (Drops to 110 HP)
Arcanine uses ExtremeSpeed
Talonflame loses 110 HP
Talonflame faints
Arcanine loses 32 HP from Life Orb
Arcanine faints
This is the average worst case scenario in this case. It assumes Talonflame has 252 EVs in HP (which is unlikely--most people run at least some speed) and Arcanine has none. It also assumes that you're switching in to Brave Bird and not Flare Blitz.
Second scenario with Wild Charge coming next post.