Parametric, Polar and Vector-Valued Functions · Superchallenge problem

For $a>1$, let $A(a)$ be the area enclosed by the limaçon $r=a+\cos\theta$. As the limaçon grows huge, does its area come to behave like that of a plain circle of radius $a$? Find $\displaystyle\lim_{a\to\infty}\dfrac{A(a)}{\pi a^2}$.