animals developing limbs?

Okay, I have always wondered something about evolution. How is it possible through natural selection for the amphibians and other species that grew limbs and crawled out of the water to become land animals to have done so? It just doesn't make sense. In natural selection an organism will have a new trait that is advantageous, and then reproduce and more will have that trait, and then the one's that do not have the trait will die off, and then that species has evolved. It makes sense for speed and strength, but not for the growth of limbs...especially since it took place over millions of years, the first little change would be a small stub at the very least, and that would not help the animal at all, much less cause it to evolve further...Can anybody answer this?

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