What's the best way to face a customer who's using bad words for a small mistake?

First thing you must to do is listening without saying any word "Any word you will say well let him more angry whatever the word was".
Look directly to his eyes and don't let him feel that you do not give attention to his complain.
Let your look be very polite NOT provocatively, And sure without laughing "Eye Contact" Try to ignore the bad words, Small mistake to you maybe a BIG one to your customer.
After he finish, Let him feel that you're understanding why he must be angry, Be careful to not deliver "The problem is small" message to him, If you did that, CONGRATULATIONS you put fire beside gas, So be ready to the fire.
If your customer feel that you're interested in the problem, He will feel better and give a space to find a solution with you, Sure no problem to apologize for this mistake.

For a successful service delivery and to make an angry customer in to happy one, try focusing on some of the most essential successful customer service delivery aspects including: a) Never argue back: b) Use your ears more than your mouth: c) Be patient: d) Show that you care: e) Be positive in your approach f) Control your anger: g) De-stress yourself from time to time:

keep calm , carefully smile , ask the customer to have a seat in the customer's waiting area and try to listen only , speak when you have a solution .

Never argue back but listening, being empathic, being patient, being positive in your approach and know that is not against you, controlling your anger and trying to be his adviser and guide.

Listening without interrupting always cools off the other person upto some extend.
Acknowledge mistake if you know it was a mistake once the customer is done with the complaint & offer apology very politely.
offer assurance that you will personally ensure to resolved the matter for him / her.
Tell customer to give you a few minutes to look into the matter to buy you time & meanwhile he/she also will calm down & will feel you are really going out of your way to help him/her & will thank you later.
Ensure you also find the best option for the problem & inform the customer without delay.

Put a smile on your face and apologize to the customer and listen to his issue and solve the problem on the spot.
I had same experince and at the end the customer didnt stop aplogizing for the bad words that he used.
Always make sure to feel the customer that he is right

Handle him as following: - Don't interrupt.
- Don't take the criticism (judgment) personally.
- keep irate customers out of the public eye.
- Don't blame.
- Don't argue.
- Be warm and friendly.
- Be professional.
- No false promises.
You should ask the customer directly; what seems to be the problem? What has caused this problem (if it's not obvious)? How can I rectify the problem, how can Ii make it better? You need to deal with the complaint quickly, quietly and without arguments Sometimes you get customers who are just itching for a fight or a loud argument Never, ever, engage in that sort of activity.
Never get angry or show any signs of frustration.
This is counter productive since it will only make them more angry that you're arguing back and if they create a scene in front of other customers the dissatisfaction could easily spread creating more complaints to deal with.
Make lots of eye contact, that shows that you're really paying attention to the customer's grievance .Apologize, if you sure that the complain is correct,do something about it and Follow up,

Try to get information about what is his/her problem.
Comfort him/her.
Find a very fast and professional soulation for his/her problem.
You've got to be a good listener.Have patience to help them to overcome their angre.

Pay full attention to the customer and take down notes if needed about his points that he delivers. make feel him you are going to solve it  then or later as you take down notes. After he finishes,lay emphasize on the actions that you are going to take to solve the problem. Dont say no or any negetivity for the first answer. If it is the case to be handled by the manager and if he is not available at moment just try for him as you do normally  and inform the customer that he will be available shortly.

1-Put yourself in his/her shoes 2-Do not say a word for at least 2 minutes 3-Assure him/her that issue will end or start ending today with you.
4-give him your name and ID if you had to, bring a paper and a pen and let him/her write down the current situation in points.
5-while solving the issue let him write down the situation after getting resolved, make a comparison between the issue before your interference and after.
Difference should be noticed, if not, make sure that you reapply another solution.
until he/she notice the difference.

Listen to the end, and go direct to the point with best solution...