One argument for Limited Atonement is that if Jesus died for everyone, and yet not everyone will be saved, that means Jesus failed. If even one person whom God wants to be saved is not saved, God must have been too weak to save that person.
But here's the problem: The Bible repeatedly affirms that God wants everyone to be saved. Indeed, Calvinists repeatedly have to reinterpret words like "all", "any", and "world" whenever it seems to suggest that God wants all men to be saved (1 Timothy 2:4), that He is not willing that any should perish (2 Peter 3:9), or that Jesus is the propitiation for the sins of the whole world (1 John 2:2).
As it happens, there is another place in the Bible where the word "world" is mentioned that solves the whole "Jesus failed" argument. You've likely heard this verse over and over again. Maybe it was even preached to you before you were saved. In John 3:16, we are told that God so loved the world that He gave His only Son. But it doesn't say "so that everyone will be saved". The purpose of God giving us His Son is so that whoever believes in Him would not perish, but have eternal life.
Now, according to any non-Calvinist interpretation, did Jesus fail if He wants someone saved, but they don't get saved? I would say absolutely not. We're talking the difference between opening a door and forcing someone through it. If Jesus died so that whoever believes in Him would not die, but have everlasting life, you do not have to be a Calvinist to say He succeeded, because everyone who believes in Him will have eternal life. Jesus' mission was a glorious success, and one which we should all be grateful to benefit from.