The OFFICIAL age limit to play in LCS is 17.
You can play in CS at 16, but to compete in the playoffs you have to celebrate your 17th birthday before the start of the next LCS split (the last thing is not explicitly written in the ruleset, but the teams that get promoted from CS into LCS must play with at least 3 members of the active roster that has been promoted to LCS in the first game of the split, so they generally tend to bench underage players during playoffs, players that would not be eligilble for LCS at the start of the subsequent split)
"1.1
Player and Team Eligibility
No team shall be able to compete in any CS match without a minimum of three starting players who would potentially be able to compete in the next LCS split. To be considered potentially eligible to compete in the next LCS split, a CS player must have lived 17 full years (by or before the start of the next LCS split) and may not have any current disciplinary suspension from the LCS that would prevent him/her from playing in the first game of the season. Teams are allowed a maximum of two players on their roster who have not lived 17 full years (by or before the start of the next LCS split). All CS players must have lived 16 full years
to compete at any time during the Challenger Series. "
https://riot-web-static.s3.amazonaws.com/lolesports/Rule%20Sets/2016_Spring_EUCS_Rule_Set_V3_01.pdf
"1.1
Player Age
No player shall be considered eligible to participate in any LCS - affiliated match before his or her 17th birthday, defined as having lived 17 full years. This shall not, however, prevent teams from signing Free Agents who are 16 years old, granted that they may not participate in an LCS match until they turn 17."
https://riot-web-static.s3.amazonaws.com/lolesports/Rule%20Sets/2016%20Spring%20LCS%20Rule%20Setv3_01.pdf