Calculate this awesome limit!

Calculus Level 4

Suppose that circles of equal diameter are packed tightly in \(n\) rows inside an equilateral triangle. If \(A\) is the area of the triangle and \(B\) is the total area occupied by the \(n\) rows of circles, then find the limit of the ratio \(\frac{B}{A}\) (correct to 3 decimal places) as the value of \(n\) approaches infinity.

The above figure shows the case for \(n = 4\)

*Courtesy : Stewart Calculus Early Transcendentals Sixth Edition *

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