You would not be able to trust the results if more than 1 record is returned since the data should be specific to 1 player so I do not recommend using the LIMIT 1 statement.
Instead of using name, use steam id as this field is unique to a player and you will only get 1 result.
Code:
SELECT csgo_clans_applications.name, rcsgo_player.points, csgo_ranks.rank
FROM (csgo_clans_applications
LEFT JOIN rcsgo_player
ON csgo_clans_applications.steamid = rcsgo_player.steamid)
LEFT JOIN csgo_ranks
ON csgo_clans_applications.steamid = csgo_ranks.steamid
WHERE clan = '%i'
ORDER BY rank DESC, points DESC
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