Sivir IS a killing machine. Just not a 1 on 1 killing machine. Most of your Kills and Assists come from team fights where you need to position properly. You want to be protected from disables and AoEs while being able to get bounces going. Once the AoEs and stuns have flown then you can move into a more aggressive position and start picking up the kills.
It's not at all uncommon for a good Sivir to walk away from a teamfight with a double, triple, or quad kill and the rest in assists.
Sivir with ult will put out a lot of damage. Her short range does demand a more conservative build though. For example a typical mid/end-ish build with her might be: Stark's, Beserker Greeves, Infinity Edge, BF Sword, Banshee's, Guardian Angel.
Sivir's strength also comes in her farm. What other carries have at the 40:00 mark you should have at the 25 or 30. If you aren't getting 200+ creep kills per game you're doing it wrong (or winning very very fast).
For 1:1 Sivir can still be decent though. You just need patience and timing.
Chase/Kite with attack -> move -> attack. Don't just auto-attack. You want that dodge to be working for you. Use Boomerang when you have them stationary or in a forced path. Don't blow it too early and have them dodge it. Also spell shield is really important. You need to be able to activate it after they cast their projectile. If you can't time it that accurately you're going to fail a lot as Sivir.
Avoid some champions. There are some champions Sivir just has no business fighting 1v1. Champs like Yi, Shaco and most other melee phyiscal DPS will eat you up. Likewise burst champs like Veiger can ruin your day. Just avoid these guys 1v1 and use your team to take them down.
Solo can help Sivir a lot too. She doesn't need protecting early on and her solo gets her farm in full swing faster + gives you better ult and bounces for team fights early on.
For my build (well got it from someone else) I typically do:
Teleport + Rally/heal
Meki Pendant -> Philospher's (if I can get <6min) OR Chalice if >6min
Boots -> Besekers (hold on besekers if you can afford BF first or need other things)
then it's a bit game dependant on the ordering but the next 'set' of items are:
Beseker's Greeves (if not already present)
BF sword x 2
Emblem of Valour
Health Crystal -> Catalyst
Basically if you need the health and regen get Health Crystal + Valour first. Otherwise nab the BF sword(s) then get the health crystal/Valour.
After 25min sell Philospher's when needed. If you have chalice sell it after you have Banshee's or Catalyst (hold on to it if they're strong Magic damage)
Then we start completing items:
Then comes the late game item completions:
The 'final' set is:
Starks, Infinity Edge, Beseker Greeves, Banshee's Veil, Black Cleaver, Guardian Angel
GA is really important on Sivir late game. With the short range you can't fully avoid being targeted. Having GA gives you a ton of survivability and the ability to pop up after everything has been blown.
Likewise Banshee's gives you incredibly survivability mid game and lets you withstand a lot of punishment before you even start taking damage.
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