Christ's atonement and righteousness guarantees salvation to all for whom He served as the Substitute, that God might be just

"The sin of Adam did not make the condemnation of all men merely possible; it was the ground of their actual condemnation. So the righteousness of Christ did not make the salvation of men merely possible, it secured the actual salvation of those for whom He wrought." -Charles Hodge

"If Christ has died for you, you can never be lost. God will not punish twice for one thing. If God punished Christ for your sins He will not punish you. 'Payment God’s justice cannot twice demand; first, at the bleeding Saviour’s hand, and then again at mine.' How can God be just if he punished Christ, the substitute, and then man himself afterwards?" -C.H. Spurgeon

(Both of these quotes are from Lorraine Boettner's Limited Atonement.)

These quotes regarding the atonement answer to the questions that the doctrine of justification seeks to answer: "How can a man be just with God?" and "How can God be just but yet justify the wicked?" Only personal, perfect, and perpetual obedience will suffice; we must be dressed in spotless raiment, untainted in any respect by the defilement of totally depraved flesh. In our case as sinners, in order to be judicially declared "righteous" before God, we must obtain pardon as well as acceptance through perfect, perpetual law-keeping, as well as an infinite sacrifice for our sins -- both of which are impossible for us to perform. Because Christ served as our substitute in His passive obedience -- taking the guilt, curse, and penalty for sin upon Himself through imputation -- God has justly punished sin once and for all for His elect. Christ has also purchased the eternal reward of glorified life for His elect by His active obedience, His lifetime of perfect and perpetual Law-keeping from start to finish -- fulfilling all righteousness -- that is imputed to us. The question now becomes: "How can God be just if He punishes a man, for whom Christ died, with an eternity of hell apart from glorified life with God?"

Christ died that God might be just and the justifier of those who believe in Jesus (Rom. 3:26). Those who, by the instrumentality of faith alone, receive and rest in the person and work of Christ alone -- His active and passive obedience -- are pardoned and accepted, declared righteous before God's throne, and will obtain the eternal reward of glorified life that Christ purchased. God would be unjust to twice punish sin, first in the person of Christ through imputation of man's guilt when Christ bore their sins upon the tree, and secondly in the punishment of the man himself for the same sin. Arminianism and all other deviant forms of soteriology make God out to be an unjust judge. Either God is supposed to pardon and accept men on grounds other than an infinite sacrifice to pay the penalty for sins, as well as perfect, perpetual, and personal Law-keeping in every word, thought, deed, and motive; or He is supposed to exact the penalty for sin both from Christ and from some of those for whom He was supposed to die.