Well.... Let's put everything as it is suppose to be done.
1. What type of sports? (chess is also a sport)
2. What type of lightning (indoor/outdoor)
3. The speed and size of objects - chess vs. car rally vs. Bobsleigh
What I can say from my incredibelly small experience is that my sigma 70-300 APO DG Macro will work on some events..... but for indoor gymnastics it is not enough...
By sneaking into others
Photos were made mostly by very fast lenses:
or here You see:
But in general, phographers were simply close enough and used some standard lenses:
Of course, with some exceptions, like this guy:
that made this photo:
(exif available)
Bu hey.... somehow I canot afford to invest into lenses that cost as much as my car
so I am being left with photos like this (almost no PP done, I am too lazy now)
Sorry for flooding.
But anyway, you have always to consider plenty of different variables, and only then decide which lenses to buy. Are you going to be able be close to the subject or not (I was in the furthest available point at that gym event in Moscow)
Cheers!!