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Rationals Between

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

I am searching for fractions with values between $\sqrt{56}$ and $\sqrt{58}$. I'm told there are an infinite number of possibilities...

Find some fractions with small denominators, say less than $10$ or $20$.
Have you found any denominators which won't yield a fraction between $\sqrt{56}$ and $\sqrt{58}$?

Send us any examples you find of denominators which don't work, and a justification that all other denominators will work!