Can we find a good rational approximation for $\sqrt{5}$?

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Prove that, if \(x\) is so small that its cube and higher powers can be neglected, \[ \sqrt{\frac{1+x}{1-x}} = 1 + x + \frac{x^2}{2}. \] By taking \(x = \dfrac{1}{9}\), prove that \(\sqrt{5}\) is approximately equal to \(\dfrac{181}{81}\).

Can we rewrite the left-hand side as a product of functions that we do know how to expand?