How to know if a friend request via Facebook has been denied?

Facebook respects people's privacy, so they don't notify when friend requests are deleted or rejected. Same thing happens when you unfriend a friend from your list. Facebook only notifies when you accept them.

The only way to know that you have unfriended them is, they won't get to see the tab "Friends" when they visit your profile. Rather they would see "Send a friend request" tab replacing it.

They can also come to know if you have a common friend where your name will not appear within the "mutual friend"'s list.

It's easy to figure out if a person denied your friend request. You just have to go back to their profile. And you can see add friend button if they deleted your request. If your friend clicked on "delete request" then "Add friend" option appears back again.

If your friend clicked on "Ignore" then that "Add friend" button option remain hidden. So you may not be able to add again until that user specifically decides to do so.

This behavior is not consistent on all continents and depends on the privacy rules set by the respective countries. Countries in European union have stricter guidelines on privacy.

The privacy options can be made more strict with change in language. For example for "English (UK)"it gets more privacy as European union have stricter guidelines on privacy. If you set the language as "English(US) then the option are relaxed and you may get plenty of hints when someone ignores your friend request or clicks on option of "you don't know the person".

No, you can safely decide to ignore the friend request. The request will be shown as pending when they view your profile. But, since the request is already pending, they will not be able to send you another friend request.
However, you can decide to decline the friend request if you are sure that you would not wish to be in their friend list. Doing so will not let them know that you have declined the request, but they will be able to send you another request.

Thus Facebook does not explicitly notify the member concerned about the friend request being deleted. But it is quite easy to figure it out. So I would conclude that your friend will definitely get to know that you have deleted her request.