Falling in love at first sight is indeed a believable situation; however, it's doubtful that it's real love. It is more a feeling of infatuation, which is a "foolish or all-absorbing passion," rather than real love (Random House Dictionary). Real love is a matter of choice rather than just a feeling. Real love is felt when you truly get to know the person and decide to accept the good with the bad. What Romeo and Juliet felt when they first met, which they deemed as love at first sight, was really more of an intense, passionate physical attraction. However, as the play progresses, Juliet does mature enough to feel real love for Romeo.
Romeo's love at first sight is most clearly seen as being infatuation the moment he lays eyes on her. Romeo very clearly mistakes real love for physical attraction; however, it's perfectly believable that Romeo thought he felt real love at first sight, as young men frequently mistake love for physical attraction. We especially see that Romeo has mistaken love for physical attraction when he says, "Did my heart love till now? Forswear it, sight! / For I ne'er saw true beauty till this night" (I.v.54-55). But since Romeo clearly believes he has fallen in love, Shakespeare portrays falling in love as a believable situation.
While Juliet clearly felt equal physical attraction for Romeo, her love does progress to real love the moment she learns that Romeo has killed her cousin Tybalt. Juliet is faced with making some difficult decisions, typical of the decisions people who are in a real-love relationship must face: (1) Should she hate him or continue to love him? (2) Should she distrust him or continue to trust him? At first, her negative emotions prevail, and she feels very deceived by him, as we see in her string of oxymora:
O serpent heart, hid with a flow'ring face!
Did ever dragon keep so fair a cave?
Beautiful tyrant! Fiend angelical!
Just opposite to what thou justly seem'st--. (III.ii.76-81)
These oxymora make it very clear that Juliet used Romeo's handsome looks to judge his character, making herself believe she was really in love with him when in actuality she knew little about him. Now that it seems he has done a treacherous deed, she is faced with the possibility that he is not what he seemed to be. For example, she describes him as both a serpent and a person with a "flow'ring face." The serpent is typically seen as a symbol of evil and sin, and being "hid with a flowr'ing face" refers to his beauty. Therefore, she is arguing that his beauty deceived her, and his face actually hides an evil heart. However, Juliet does not continue to see him as deceptive. As the scene progresses, she reasons herself into seeing that Romeo must have killed Tybalt out of self-defense. She further decides that since he is her husband, she must continue to trust and honor him, showing us that the infatuation she felt for Romeo at the beginning has now blossomed into real love.