Why does $\,\lim\limits_{b\to \infty}\left(\sqrt{c}-b\right)=\frac{a}{2}\;$ when $\;c=b^2+ab\;?$

I've been studying the properties of rectangles for a little while on my own so I don't know what are the actual terms or what are the formulas, but I've noticed that if you take $A$ as a "degree of rectangleness" (so if a rectangle is $b×m=c$ and $|m-b|=A$ then the bigger $b$ is, the closer $\sqrt{c}-b$ is to $\frac{a}{2}$

Can someone explain why it works and if there's a function or formula that can explain this phenomenon?

Why does $\,\lim\limits_{b\to \infty}\left(\sqrt{c}-b\right)=\frac{a}{2}\;$ when $\;c=b^2+ab\;?$

I've been studying the properties of rectangles for a little while on my own so I don't know what are the actual terms or what are the formulas, but I've noticed that if you take $A$ as a "degree of rectangleness" (so if a rectangle is $\;b\!\cdot\!m=c\;$ and $|m\!-\!b|\!=\!A\,$) , then the bigger $b$ is, the closer $\sqrt{c}\!-\!b$ is to $\frac{A}{2}$.

Can someone explain why it works and if there's a function or formula that can explain this phenomenon?

I've been studying the properties of rectangles for a little while on my own so I don't know what are the actual terms or what are the formulas, but I've noticed that if you take $A$ as a "degree of rectangleness" (so if a rectangle is $b×m=c$ and $|m-b|=A$ then the bigger $b$ is, the closer $\sqrt{c}-b$ is to $\frac{a}{2}$

Can someone explain why it works and if there's a function or formula that can explain this phenomenon?

Why does $\,\lim\limits_{b\to \infty}\left(\sqrt{c}-b\right)=\frac{a}{2}\;$ when $\;c=b^2+ab\;?$

I've been studying the properties of rectangles for a little while on my own so I don't know what are the actual terms or what are the formulas, but I've noticed that if you take $A$ as a "degree of rectangleness" (so if a rectangle is $b×m=c$ and $|m-b|=A$ then the bigger $b$ is, the closer $\sqrt{c}-b$ is to $\frac{a}{2}$

Can someone explain why it works and if there's a function or formula that can explain this phenomenon?